Integrand size = 28, antiderivative size = 550 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d \left (d+c^2 d x^2\right )^{3/2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {7 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \] Output:
7/12*b^2*c^4*d^2*x*(c^2*d*x^2+d)^(1/2)-1/3*b^2*c^2*d*(c^2*d*x^2+d)^(3/2)/x -23/12*b^2*c^3*d^2*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)-5/2* b*c^5*d^2*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+7/3 *b*c^3*d^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))-1/3*b* c*d^2*(c^2*x^2+1)^(3/2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/x^2+5/2*c^4 *d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2-7/3*c^3*d^2*(c^2*d*x^2+d)^ (1/2)*(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2)-5/3*c^2*d*(c^2*d*x^2+d)^(3/2) *(a+b*arcsinh(c*x))^2/x-1/3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3+5 /6*c^3*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^3/b/(c^2*x^2+1)^(1/2)+14 /3*b*c^3*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)+7/3*b^2*c^3*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 = 2.03 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d^2 \left (-8 a b c x \sqrt {d+c^2 d x^2}-8 a^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-56 a^2 c^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-8 b^2 c^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+12 a^2 c^4 x^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+20 b^2 c^3 x^3 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3-6 a b c^3 x^3 \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))+112 a b c^3 x^3 \sqrt {d+c^2 d x^2} \log (c x)+60 a^2 c^3 \sqrt {d} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-56 b^2 c^3 x^3 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+3 b^2 c^3 x^3 \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))-2 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \left (4 b c x+8 a \sqrt {1+c^2 x^2}+56 a c^2 x^2 \sqrt {1+c^2 x^2}+3 b c^3 x^3 \cosh (2 \text {arcsinh}(c x))-56 b c^3 x^3 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-6 a c^3 x^3 \sinh (2 \text {arcsinh}(c x))\right )+2 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 \left (30 a c^3 x^3-4 b \left (-7 c^3 x^3+\sqrt {1+c^2 x^2}+7 c^2 x^2 \sqrt {1+c^2 x^2}\right )+3 b c^3 x^3 \sinh (2 \text {arcsinh}(c x))\right )\right )}{24 x^3 \sqrt {1+c^2 x^2}} \] Input:
Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^4,x]
Output:
(d^2*(-8*a*b*c*x*Sqrt[d + c^2*d*x^2] - 8*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^ 2*d*x^2] - 56*a^2*c^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 8*b^2*c^ 2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 12*a^2*c^4*x^4*Sqrt[1 + c^2* x^2]*Sqrt[d + c^2*d*x^2] + 20*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x] ^3 - 6*a*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] + 112*a*b*c^3* x^3*Sqrt[d + c^2*d*x^2]*Log[c*x] + 60*a^2*c^3*Sqrt[d]*x^3*Sqrt[1 + c^2*x^2 ]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 56*b^2*c^3*x^3*Sqrt[d + c^2*d *x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])] + 3*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]* Sinh[2*ArcSinh[c*x]] - 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(4*b*c*x + 8*a *Sqrt[1 + c^2*x^2] + 56*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 3*b*c^3*x^3*Cosh[2*A rcSinh[c*x]] - 56*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])] - 6*a*c^3*x^3*Sin h[2*ArcSinh[c*x]]) + 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(30*a*c^3*x^3 - 4*b*(-7*c^3*x^3 + Sqrt[1 + c^2*x^2] + 7*c^2*x^2*Sqrt[1 + c^2*x^2]) + 3*b *c^3*x^3*Sinh[2*ArcSinh[c*x]])))/(24*x^3*Sqrt[1 + c^2*x^2])
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.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x^3}dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6217 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c \int \frac {\left (c^2 x^2+1\right )^{3/2}}{x^2}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c \left (3 c^2 \int \sqrt {c^2 x^2+1}dx-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6216 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \int \sqrt {c^2 x^2+1}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {\int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (-\frac {\int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (-\frac {\int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{3} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2}dx+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {2 b c d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \left (-\frac {b c \sqrt {c^2 d x^2+d} \int x (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {2 b c d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {2 b c d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{3} c^2 d \left (3 c^2 d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {2 b c d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6216 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \int \sqrt {c^2 x^2+1}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {\int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {\int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {\int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {i \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+3 c^2 d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
Input:
Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^4,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(2296\) vs. \(2(500)=1000\).
Time = 1.24 (sec) , antiderivative size = 2297, normalized size of antiderivative = 4.18
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2297\) |
| parts | \(\text {Expression too large to display}\) | \(2297\) |
Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2/x^4,x,method=_RETURNVERBOSE)
Output:
-23/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x/(c^2*x^2+1 )*arcsinh(x*c)^2*c^2-7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2* x^2+1)*x/(c^2*x^2+1)*arcsinh(x*c)*c^4+147*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(6 3*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*c^7+35*b^2*(d *(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*ar csinh(x*c)^2*c^5-21*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1 )*x^2/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^5-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2 /(63*c^4*x^4+15*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c+21*b^2*(d* (c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7 +5*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1) ^(1/2)*c^5+49/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^ 3*arcsinh(x*c)*c^6+7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^ 2+1)*x*arcsinh(x*c)*c^4+1/2*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^6/(c^2*x^2+1)* arcsinh(x*c)^2*x^3+1/2*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^4/(c^2*x^2+1)*arcsi nh(x*c)^2*x+14/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)* ln(1-x*c-(c^2*x^2+1)^(1/2))*d^2*c^3-56/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63 *c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-71/3*b^2*(d*(c^2*x^2+1))^(1/2)* d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-16/3*b^2*(d*(c^2*x^2+1)) ^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*c^4+5/2*a^2*c^4*d^2*x*( c^2*d*x^2+d)^(1/2)+5/2*a^2*c^4*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+...
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,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^4, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \] Input:
integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2/x**4,x)
Output:
Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2/x**4, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxim a")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,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. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^4} \,d x \] Input:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^4,x)
Output:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^4, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {\sqrt {d}\, d^{2} \left (6 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-28 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-4 \sqrt {c^{2} x^{2}+1}\, a^{2}+24 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}+48 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) a b \,c^{2} x^{3}+12 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}+24 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}+24 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) a b \,c^{4} x^{3}+12 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c^{4} x^{3}+30 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2} c^{3} x^{3}+5 a^{2} c^{3} x^{3}\right )}{12 x^{3}} \] Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^2/x^4,x)
Output:
(sqrt(d)*d**2*(6*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 - 28*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - 4*sqrt(c**2*x**2 + 1)*a**2 + 24*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**4,x)*a*b*x**3 + 48*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/ x**2,x)*a*b*c**2*x**3 + 12*int((sqrt(c**2*x**2 + 1)*asinh(c*x)**2)/x**4,x) *b**2*x**3 + 24*int((sqrt(c**2*x**2 + 1)*asinh(c*x)**2)/x**2,x)*b**2*c**2* x**3 + 24*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*a*b*c**4*x**3 + 12*int(sqr t(c**2*x**2 + 1)*asinh(c*x)**2,x)*b**2*c**4*x**3 + 30*log(sqrt(c**2*x**2 + 1) + c*x)*a**2*c**3*x**3 + 5*a**2*c**3*x**3))/(12*x**3)