Integrand size = 28, antiderivative size = 519 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {31}{64} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d x \left (d+c^2 d x^2\right )^{3/2}-\frac {89 b^2 c d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 \sqrt {1+c^2 x^2}}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}+\frac {2 b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \] Output:
31/64*b^2*c^2*d^2*x*(c^2*d*x^2+d)^(1/2)+1/32*b^2*c^2*d*x*(c^2*d*x^2+d)^(3/ 2)-89/64*b^2*c*d^2*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)-15/8 *b*c^3*d^2*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+b* c*d^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))-1/8*b*c*d^2 *(c^2*x^2+1)^(3/2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+15/8*c^2*d^2*x*( c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2-c*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arc sinh(c*x))^2/(c^2*x^2+1)^(1/2)+5/4*c^2*d*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsin h(c*x))^2-(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x+5/8*c*d^2*(c^2*d*x^2+ d)^(1/2)*(a+b*arcsinh(c*x))^3/b/(c^2*x^2+1)^(1/2)+2*b*c*d^2*(c^2*d*x^2+d)^ (1/2)*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))^2)/(c^2*x^2+1)^(1/2) +b^2*c*d^2*(c^2*d*x^2+d)^(1/2)*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)/(c^2*x ^2+1)^(1/2)
Time = 2.62 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {d^2 \left (-256 a^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+288 a^2 c^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+64 a^2 c^4 x^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+160 b^2 c x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3-128 a b c x \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-4 a b c x \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))+512 a b c x \sqrt {d+c^2 d x^2} \log (c x)+480 a^2 c \sqrt {d} x \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-256 b^2 c x \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+64 b^2 c x \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+b^2 c x \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))-4 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \left (128 a \sqrt {1+c^2 x^2}+32 b c x \cosh (2 \text {arcsinh}(c x))+b c x \cosh (4 \text {arcsinh}(c x))-128 b c x \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-64 a c x \sinh (2 \text {arcsinh}(c x))-4 a c x \sinh (4 \text {arcsinh}(c x))\right )+8 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 \left (60 a c x+32 b c x-32 b \sqrt {1+c^2 x^2}+16 b c x \sinh (2 \text {arcsinh}(c x))+b c x \sinh (4 \text {arcsinh}(c x))\right )\right )}{256 x \sqrt {1+c^2 x^2}} \] Input:
Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^2,x]
Output:
(d^2*(-256*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 288*a^2*c^2*x^2*Sqr t[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 64*a^2*c^4*x^4*Sqrt[1 + c^2*x^2]*Sqrt [d + c^2*d*x^2] + 160*b^2*c*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 128*a*b *c*x*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 4*a*b*c*x*Sqrt[d + c^2*d*x ^2]*Cosh[4*ArcSinh[c*x]] + 512*a*b*c*x*Sqrt[d + c^2*d*x^2]*Log[c*x] + 480* a^2*c*Sqrt[d]*x*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 256*b^2*c*x*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])] + 64*b^ 2*c*x*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + b^2*c*x*Sqrt[d + c^2*d*x^ 2]*Sinh[4*ArcSinh[c*x]] - 4*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(128*a*Sqrt [1 + c^2*x^2] + 32*b*c*x*Cosh[2*ArcSinh[c*x]] + b*c*x*Cosh[4*ArcSinh[c*x]] - 128*b*c*x*Log[1 - E^(-2*ArcSinh[c*x])] - 64*a*c*x*Sinh[2*ArcSinh[c*x]] - 4*a*c*x*Sinh[4*ArcSinh[c*x]]) + 8*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*( 60*a*c*x + 32*b*c*x - 32*b*Sqrt[1 + c^2*x^2] + 16*b*c*x*Sinh[2*ArcSinh[c*x ]] + b*c*x*Sinh[4*ArcSinh[c*x]])))/(256*x*Sqrt[1 + c^2*x^2])
Result contains complex when optimal does not.
Time = 4.79 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.20, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6222, 6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 222, 6216, 211, 211, 222, 6216, 211, 222, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \int \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \int x (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{3/2}dx}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+5 c^2 d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 6216 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{4} b c \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{4} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 6216 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {\int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {\int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {\int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {i \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+5 c^2 d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
Input:
Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^2,x]
Output:
-(((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x) + 5*c^2*d*((x*(d + c^2 *d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*d*((x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b *c*Sqrt[1 + c^2*x^2]) - (b*c*Sqrt[d + c^2*d*x^2]*((x^2*(a + b*ArcSinh[c*x] ))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/2))/Sq rt[1 + c^2*x^2]))/4 - (b*c*d*Sqrt[d + c^2*d*x^2]*(((1 + c^2*x^2)^2*(a + b* ArcSinh[c*x]))/(4*c^2) - (b*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 + c ^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 + c^2*x^2])) + (2* b*c*d^2*Sqrt[d + c^2*d*x^2]*(((1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/2 + ((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/4 - (b*c*((x*Sqrt[1 + c^2*x^2])/2 + Arc Sinh[c*x]/(2*c)))/2 - (b*c*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 + c^ 2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/4 + (I*((-1/2*I)*(a + b*ArcSinh[c*x]) ^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c*x])*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c*x]))/b)]) + (b^2*PolyLog[2, -a - b*ArcSinh[c*x]])/4)))/b))/ Sqrt[1 + c^2*x^2]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSinh[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcSinh[c*x])/x), x], x] - Simp[b*c*(d^p /(2*p)) Int[(1 + c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x ^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e , f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Time = 1.36 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}+\frac {5 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2} c^{2} d x}{4}+\frac {15 a^{2} d^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2} x}{8}+\frac {15 a^{2} c^{2} d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+2 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+72 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-72 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+33 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+40 \operatorname {arcsinh}\left (x c \right )^{3} x c +128 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x c +128 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) x c -64 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) x c +128 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) x c -64 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2}-33 x c \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 x^{5} c^{5}+144 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-72 x^{3} c^{3}+120 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 x c \,\operatorname {arcsinh}\left (x c \right )-128 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-33 x c \right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}\) | \(589\) |
| parts | \(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}+\frac {5 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2} c^{2} d x}{4}+\frac {15 a^{2} d^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2} x}{8}+\frac {15 a^{2} c^{2} d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+2 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+72 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-72 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+33 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+40 \operatorname {arcsinh}\left (x c \right )^{3} x c +128 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x c +128 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) x c -64 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) x c +128 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) x c -64 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2}-33 x c \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 x^{5} c^{5}+144 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-72 x^{3} c^{3}+120 \operatorname {arcsinh}\left (x c \right )^{2} x c +128 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 x c \,\operatorname {arcsinh}\left (x c \right )-128 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-33 x c \right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}\) | \(589\) |
Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2/x^2,x,method=_RETURNVERBOSE)
Output:
-a^2/d/x*(c^2*d*x^2+d)^(7/2)+a^2*c^2*x*(c^2*d*x^2+d)^(5/2)+5/4*(c^2*d*x^2+ d)^(3/2)*a^2*c^2*d*x+15/8*a^2*d^2*(c^2*d*x^2+d)^(1/2)*c^2*x+15/8*a^2*c^2*d ^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/64*b^2*(d *(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x*(16*(c^2*x^2+1)^(1/2)*arcsinh(x*c) ^2*x^4*c^4-8*arcsinh(x*c)*x^5*c^5+2*x^4*c^4*(c^2*x^2+1)^(1/2)+72*arcsinh(x *c)^2*(c^2*x^2+1)^(1/2)*x^2*c^2-72*arcsinh(x*c)*x^3*c^3+33*x^2*c^2*(c^2*x^ 2+1)^(1/2)+40*arcsinh(x*c)^3*x*c+128*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/ 2))*x*c+128*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*x*c-64*arcsinh(x*c)^2 *x*c+128*polylog(2,x*c+(c^2*x^2+1)^(1/2))*x*c+128*polylog(2,-x*c-(c^2*x^2+ 1)^(1/2))*x*c-64*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2-33*x*c*arcsinh(x*c))*d^2 +1/64*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x*(32*arcsinh(x*c)*(c^2* x^2+1)^(1/2)*x^4*c^4-8*x^5*c^5+144*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2- 72*x^3*c^3+120*arcsinh(x*c)^2*x*c+128*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)*x*c- 128*x*c*arcsinh(x*c)-128*arcsinh(x*c)*(c^2*x^2+1)^(1/2)-33*x*c)*d^2
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^2,x, algorithm="frica s")
Output:
integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d)/x^2, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \] Input:
integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2/x**2,x)
Output:
Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2/x**2, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^2,x, algorithm="maxim a")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^2,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^2} \,d x \] Input:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^2,x)
Output:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^2, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {\sqrt {d}\, d^{2} \left (2 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+9 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a^{2}+16 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) a b x +8 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} x +16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{4} x +32 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) a b \,c^{2} x +8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{4} x +16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c^{2} x +15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2} c x -10 a^{2} c x \right )}{8 x} \] Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^2/x^2,x)
Output:
(sqrt(d)*d**2*(2*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 + 9*sqrt(c**2*x**2 + 1 )*a**2*c**2*x**2 - 8*sqrt(c**2*x**2 + 1)*a**2 + 16*int((sqrt(c**2*x**2 + 1 )*asinh(c*x))/x**2,x)*a*b*x + 8*int((sqrt(c**2*x**2 + 1)*asinh(c*x)**2)/x* *2,x)*b**2*x + 16*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*a*b*c**4*x + 32*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*a*b*c**2*x + 8*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**2,x)*b**2*c**4*x + 16*int(sqrt(c**2*x**2 + 1)*asinh (c*x)**2,x)*b**2*c**2*x + 15*log(sqrt(c**2*x**2 + 1) + c*x)*a**2*c*x - 10* a**2*c*x))/(8*x)