Integrand size = 28, antiderivative size = 412 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {32 b^2 \sqrt {d+c^2 d x^2}}{9 c^6 d^2}+\frac {2 b^2 \left (d+c^2 d x^2\right )^{3/2}}{27 c^6 d^3}+\frac {10 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2}+\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}} \] Output:
-32/9*b^2*(c^2*d*x^2+d)^(1/2)/c^6/d^2+2/27*b^2*(c^2*d*x^2+d)^(3/2)/c^6/d^3 +10/3*b*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^5/d/(c^2*d*x^2+d)^(1/2)-2 /9*b*x^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3/d/(c^2*d*x^2+d)^(1/2)-x^ 4*(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(1/2)-8/3*(c^2*d*x^2+d)^(1/2)*( a+b*arcsinh(c*x))^2/c^6/d^2+4/3*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)) ^2/c^4/d^2+4*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1) ^(1/2))/c^6/d/(c^2*d*x^2+d)^(1/2)-2*I*b^2*(c^2*x^2+1)^(1/2)*polylog(2,-I*( c*x+(c^2*x^2+1)^(1/2)))/c^6/d/(c^2*d*x^2+d)^(1/2)+2*I*b^2*(c^2*x^2+1)^(1/2 )*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c^6/d/(c^2*d*x^2+d)^(1/2)
Time = 0.73 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-72 a^2-94 b^2-36 a^2 c^2 x^2-92 b^2 c^2 x^2+9 a^2 c^4 x^4+2 b^2 c^4 x^4+90 a b c x \sqrt {1+c^2 x^2}-6 a b c^3 x^3 \sqrt {1+c^2 x^2}-144 a b \text {arcsinh}(c x)-72 a b c^2 x^2 \text {arcsinh}(c x)+18 a b c^4 x^4 \text {arcsinh}(c x)+90 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-72 b^2 \text {arcsinh}(c x)^2-36 b^2 c^2 x^2 \text {arcsinh}(c x)^2+9 b^2 c^4 x^4 \text {arcsinh}(c x)^2+108 a b \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-54 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+54 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-54 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+54 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )}{27 c^6 d \sqrt {d+c^2 d x^2}} \] Input:
Integrate[(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]
Output:
(-72*a^2 - 94*b^2 - 36*a^2*c^2*x^2 - 92*b^2*c^2*x^2 + 9*a^2*c^4*x^4 + 2*b^ 2*c^4*x^4 + 90*a*b*c*x*Sqrt[1 + c^2*x^2] - 6*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 144*a*b*ArcSinh[c*x] - 72*a*b*c^2*x^2*ArcSinh[c*x] + 18*a*b*c^4*x^4*Arc Sinh[c*x] + 90*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^3*x^3*Sqrt [1 + c^2*x^2]*ArcSinh[c*x] - 72*b^2*ArcSinh[c*x]^2 - 36*b^2*c^2*x^2*ArcSin h[c*x]^2 + 9*b^2*c^4*x^4*ArcSinh[c*x]^2 + 108*a*b*Sqrt[1 + c^2*x^2]*ArcTan [Tanh[ArcSinh[c*x]/2]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (54*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E ^ArcSinh[c*x]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c* x]] + (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(27*c^6*d *Sqrt[d + c^2*d*x^2])
Time = 2.97 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.16, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6225, 6227, 243, 53, 2009, 6191, 243, 53, 2009, 6213, 2009, 6227, 241, 6204, 3042, 4668, 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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}+\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {4 \left (-\frac {2 b \sqrt {c^2 x^2+1} \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c \sqrt {c^2 d x^2+d}}-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}-\frac {b \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx}{3 c}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {4 \left (-\frac {2 b \sqrt {c^2 x^2+1} \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c \sqrt {c^2 d x^2+d}}-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}-\frac {b \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2}{6 c}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {4 \left (-\frac {2 b \sqrt {c^2 x^2+1} \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c \sqrt {c^2 d x^2+d}}-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}-\frac {b \int \left (\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {1}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2}{6 c}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-\frac {2 b \sqrt {c^2 x^2+1} \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c \sqrt {c^2 d x^2+d}}-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx\right )}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2\right )}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \left (\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {1}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}+\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {4 \left (-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))dx}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{c^2}-\frac {b \int \frac {x}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{c^2}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\) |
Input:
Int[(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]
Output:
-((x^4*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2])) + (4*((x^2*Sqr t[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^2*d) - (2*b*Sqrt[1 + c^2*x^2 ]*(-1/6*(b*c*((-2*Sqrt[1 + c^2*x^2])/c^4 + (2*(1 + c^2*x^2)^(3/2))/(3*c^4) )) + (x^3*(a + b*ArcSinh[c*x]))/3))/(3*c*Sqrt[d + c^2*d*x^2]) - (2*((Sqrt[ d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(c^2*d) - (2*b*Sqrt[1 + c^2*x^2]*(a *x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/(c*Sqrt[d + c^2*d*x^2])) )/(3*c^2)))/(c^2*d) + (2*b*Sqrt[1 + c^2*x^2]*(-1/6*(b*((-2*Sqrt[1 + c^2*x^ 2])/c^4 + (2*(1 + c^2*x^2)^(3/2))/(3*c^4)))/c + (x^3*(a + b*ArcSinh[c*x])) /(3*c^2) - (-((b*Sqrt[1 + c^2*x^2])/c^3) + (x*(a + b*ArcSinh[c*x]))/c^2 - (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^Arc Sinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])/c^3)/c^2))/(c*d*Sqrt[d + c^ 2*d*x^2])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 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_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 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_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(p + 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]) /; Fre eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 1.50 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.61
| method | result | size |
| default | \(a^{2} \left (\frac {x^{4}}{3 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )}{3 c^{2}}\right )+\frac {i b^{2} \left (36 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 i \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-84 i \operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}-9 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+72 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}-54 \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-54 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+54 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+54 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+92 i \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-90 i \operatorname {arcsinh}\left (x c \right ) x c +54 \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+54 \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-54 \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-54 \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+6 i \operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+94 i \sqrt {c^{2} x^{2}+1}\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{27 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2} c^{6}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-12 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+9 i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-9 i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )+15 \sqrt {c^{2} x^{2}+1}\, x c -24 \,\operatorname {arcsinh}\left (x c \right )\right )}{9 d^{2} c^{6} \left (c^{2} x^{2}+1\right )}\) | \(663\) |
| parts | \(a^{2} \left (\frac {x^{4}}{3 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )}{3 c^{2}}\right )+\frac {i b^{2} \left (36 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 i \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-84 i \operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}-9 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+72 i \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}-54 \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-54 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+54 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+54 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+92 i \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-90 i \operatorname {arcsinh}\left (x c \right ) x c +54 \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+54 \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-54 \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-54 \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+6 i \operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+94 i \sqrt {c^{2} x^{2}+1}\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{27 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2} c^{6}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-12 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+9 i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-9 i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )+15 \sqrt {c^{2} x^{2}+1}\, x c -24 \,\operatorname {arcsinh}\left (x c \right )\right )}{9 d^{2} c^{6} \left (c^{2} x^{2}+1\right )}\) | \(663\) |
Input:
int(x^5*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
a^2*(1/3*x^4/c^2/d/(c^2*d*x^2+d)^(1/2)-4/3/c^2*(x^2/c^2/d/(c^2*d*x^2+d)^(1 /2)+2/d/c^4/(c^2*d*x^2+d)^(1/2)))+1/27*I*b^2/(c^2*x^2+1)^(3/2)*(36*I*arcsi nh(x*c)^2*(c^2*x^2+1)^(1/2)*x^2*c^2-2*I*(c^2*x^2+1)^(1/2)*x^4*c^4-84*I*arc sinh(x*c)*x^3*c^3-9*I*arcsinh(x*c)^2*(c^2*x^2+1)^(1/2)*x^4*c^4+72*I*arcsin h(x*c)^2*(c^2*x^2+1)^(1/2)-54*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*arcsinh(x*c) *x^2*c^2-54*arcsinh(x*c)*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))+54*arcsinh(x*c)*l n(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2+54*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^ 2+1)^(1/2)))+92*I*(c^2*x^2+1)^(1/2)*x^2*c^2-90*I*arcsinh(x*c)*x*c+54*dilog (1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2+54*dilog(1-I*(x*c+(c^2*x^2+1)^(1/2)) )-54*dilog(1+I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2-54*dilog(1+I*(x*c+(c^2*x^2 +1)^(1/2)))+6*I*arcsinh(x*c)*x^5*c^5+94*I*(c^2*x^2+1)^(1/2))*(d*(c^2*x^2+1 ))^(1/2)/d^2/c^6+2/9*a*b*(d*(c^2*x^2+1))^(1/2)*(3*arcsinh(x*c)*c^4*x^4-(c^ 2*x^2+1)^(1/2)*c^3*x^3-12*arcsinh(x*c)*c^2*x^2+9*I*(c^2*x^2+1)^(1/2)*ln(x* c+(c^2*x^2+1)^(1/2)+I)-9*I*(c^2*x^2+1)^(1/2)*ln(x*c+(c^2*x^2+1)^(1/2)-I)+1 5*(c^2*x^2+1)^(1/2)*x*c-24*arcsinh(x*c))/d^2/c^6/(c^2*x^2+1)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x, algorithm="frica s")
Output:
integral((b^2*x^5*arcsinh(c*x)^2 + 2*a*b*x^5*arcsinh(c*x) + a^2*x^5)*sqrt( c^2*d*x^2 + d)/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**5*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(3/2),x)
Output:
Integral(x**5*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(3/2), x)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x, algorithm="maxim a")
Output:
1/3*a^2*(x^4/(sqrt(c^2*d*x^2 + d)*c^2*d) - 4*x^2/(sqrt(c^2*d*x^2 + d)*c^4* d) - 8/(sqrt(c^2*d*x^2 + d)*c^6*d)) + 1/3*(b^2*c^4*sqrt(d)*x^4 - 4*b^2*c^2 *sqrt(d)*x^2 - 8*b^2*sqrt(d))*sqrt(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 1 ))^2/(c^8*d^2*x^2 + c^6*d^2) + integrate(2/3*((4*b^2*c^3*x^3 + (3*a*b*c^5 - b^2*c^5)*x^5 + 8*b^2*c*x)*(c^2*x^2 + 1) + (3*b^2*c^4*x^4 + (3*a*b*c^6 - b^2*c^6)*x^6 + 12*b^2*c^2*x^2 + 8*b^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c ^2*x^2 + 1))/(c^10*d^(3/2)*x^5 + 2*c^8*d^(3/2)*x^3 + c^6*d^(3/2)*x + (c^9* d^(3/2)*x^4 + 2*c^7*d^(3/2)*x^2 + c^5*d^(3/2))*sqrt(c^2*x^2 + 1)), x)
Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),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. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((x^5*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(3/2),x)
Output:
int((x^5*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(3/2), x)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{8} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{6}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{8} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{6}}{3 \sqrt {d}\, c^{6} d \left (c^{2} x^{2}+1\right )} \] Input:
int(x^5*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x)
Output:
(sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 - 4*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - 8*sqrt(c**2*x**2 + 1)*a**2 + 6*int((asinh(c*x)*x**5)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**8*x**2 + 6*int((asinh(c*x)*x **5)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**6 + 3 *int((asinh(c*x)**2*x**5)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**8*x**2 + 3*int((asinh(c*x)**2*x**5)/(sqrt(c**2*x**2 + 1)* c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**6)/(3*sqrt(d)*c**6*d*(c**2*x** 2 + 1))