Integrand size = 28, antiderivative size = 687 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b c^3 x (a+b \text {arcsinh}(c x))}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d^2 x \sqrt {d+c^2 d x^2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \] Output:
1/3*b^2*c^2/d^2/(c^2*d*x^2+d)^(1/2)+1/3*b*c^3*x*(a+b*arcsinh(c*x))/d^2/(c^ 2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-b*c*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x) )/d^2/x/(c^2*d*x^2+d)^(1/2)-5/6*c^2*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^( 3/2)-1/2*(a+b*arcsinh(c*x))^2/d/x^2/(c^2*d*x^2+d)^(3/2)-5/2*c^2*(a+b*arcsi nh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)+26/3*b*c^2*(c^2*x^2+1)^(1/2)*(a+b*arcsi nh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)+5*c^2*(c^2* x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2*arctanh(c*x+(c^2*x^2+1)^(1/2))/d^2/(c^2* d*x^2+d)^(1/2)-b^2*c^2*(c^2*x^2+1)^(1/2)*arctanh((c^2*x^2+1)^(1/2))/d^2/(c ^2*d*x^2+d)^(1/2)+5*b*c^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,- c*x-(c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)-13/3*I*b^2*c^2*(c^2*x^2+1)^ (1/2)*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^2/(c^2*d*x^2+d)^(1/2)+13/3*I *b^2*c^2*(c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d^2/(c^2*d *x^2+d)^(1/2)-5*b*c^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,c*x+( c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)-5*b^2*c^2*(c^2*x^2+1)^(1/2)*poly log(3,-c*x-(c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)+5*b^2*c^2*(c^2*x^2+1 )^(1/2)*polylog(3,c*x+(c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)
Time = 7.61 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^(5/2)),x]
Output:
Sqrt[d*(1 + c^2*x^2)]*(-1/2*a^2/(d^3*x^2) - (a^2*c^2)/(3*d^3*(1 + c^2*x^2) ^2) - (2*a^2*c^2)/(d^3*(1 + c^2*x^2))) - (5*a^2*c^2*Log[x])/(2*d^(5/2)) + (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(2*d^(5/2)) + (a*b*c^2* ((4*c*x)/Sqrt[1 + c^2*x^2] - 48*ArcSinh[c*x] - (8*ArcSinh[c*x])/(1 + c^2*x ^2) + 104*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 6*Sqrt[1 + c^2* x^2]*Coth[ArcSinh[c*x]/2] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csch[ArcSinh[ c*x]/2]^2 - 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 60*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 60*Sqrt[1 + c^2*x^2]*PolyLog[2 , E^(-ArcSinh[c*x])] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/ 2]^2 + 6*Sqrt[1 + c^2*x^2]*Tanh[ArcSinh[c*x]/2]))/(12*d^2*Sqrt[d*(1 + c^2* x^2)]) + (b^2*c^2*(8 + (8*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 48*ArcSinh [c*x]^2 - (8*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 12*Sqrt[1 + c^2*x^2]*ArcSinh[ c*x]*Coth[ArcSinh[c*x]/2] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Csch[ArcSin h[c*x]/2]^2 - 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x] )] - (104*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (1 04*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 24*Sqrt[1 + c^2*x^ 2]*Log[Tanh[ArcSinh[c*x]/2]] - 120*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[ 2, -E^(-ArcSinh[c*x])] - (104*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^Ar...
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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (c^2 d x^2+d\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+b c \int \frac {1}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx^2-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {2}{\sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+\frac {2}{\sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}}dx+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {5}{2} c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{d \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 d x^2+d}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 d x^2+d}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 d x^2+d}}dx}{d}-\frac {2 b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {c^2 x^2+1} \left (-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 d x^2+d}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {c^2 x^2+1} \left (-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d \sqrt {c^2 d x^2+d}}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 d x^2+d}}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 d x^2+d}}dx}{d}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{d \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c^2 x^2+1} \int \frac {(a+b \text {arcsinh}(c x))^2}{c x}d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{d \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c^2 x^2+1} \int i (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{d \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\frac {i \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{d \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^(5/2)),x]
Output:
$Aborted
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Input:
int((a+b*arcsinh(x*c))^2/x^3/(c^2*d*x^2+d)^(5/2),x)
Output:
int((a+b*arcsinh(x*c))^2/x^3/(c^2*d*x^2+d)^(5/2),x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^ 2)/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*asinh(c*x))**2/x**3/(c**2*d*x**2+d)**(5/2),x)
Output:
Integral((a + b*asinh(c*x))**2/(x**3*(d*(c**2*x**2 + 1))**(5/2)), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
1/6*a^2*(15*c^2*arcsinh(1/(c*abs(x)))/d^(5/2) - 15*c^2/(sqrt(c^2*d*x^2 + d )*d^2) - 5*c^2/((c^2*d*x^2 + d)^(3/2)*d) - 3/((c^2*d*x^2 + d)^(3/2)*d*x^2) ) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/((c^2*d*x^2 + d)^(5/2)*x^ 3) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/((c^2*d*x^2 + d)^(5/2)*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(5/2),x, algorithm="giac" )
Output:
integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^(5/2)*x^3), x)
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:
int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)^(5/2)),x)
Output:
int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)^(5/2)), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-15 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a^{2}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) a b \,c^{4} x^{6}+24 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) a b \,c^{2} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) a b \,x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b^{2} c^{4} x^{6}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b^{2} c^{2} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b^{2} x^{2}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2} c^{6} x^{6}-30 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2} c^{4} x^{4}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2} c^{2} x^{2}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2} c^{6} x^{6}+30 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2} c^{4} x^{4}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2} c^{2} x^{2}}{6 \sqrt {d}\, d^{2} x^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/x^3/(c^2*d*x^2+d)^(5/2),x)
Output:
( - 15*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 - 20*sqrt(c**2*x**2 + 1)*a**2*c* *2*x**2 - 3*sqrt(c**2*x**2 + 1)*a**2 + 12*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**7 + 2*sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3 ),x)*a*b*c**4*x**6 + 24*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**7 + 2* sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3),x)*a*b*c**2*x**4 + 12*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**7 + 2*sqrt(c**2*x**2 + 1 )*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3),x)*a*b*x**2 + 6*int(asinh(c*x)**2/ (sqrt(c**2*x**2 + 1)*c**4*x**7 + 2*sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c* *2*x**2 + 1)*x**3),x)*b**2*c**4*x**6 + 12*int(asinh(c*x)**2/(sqrt(c**2*x** 2 + 1)*c**4*x**7 + 2*sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x **3),x)*b**2*c**2*x**4 + 6*int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**4*x** 7 + 2*sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3),x)*b**2*x* *2 - 15*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a**2*c**6*x**6 - 30*log(sqrt(c* *2*x**2 + 1) + c*x - 1)*a**2*c**4*x**4 - 15*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a**2*c**2*x**2 + 15*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a**2*c**6*x**6 + 30*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a**2*c**4*x**4 + 15*log(sqrt(c**2 *x**2 + 1) + c*x + 1)*a**2*c**2*x**2)/(6*sqrt(d)*d**2*x**2*(c**4*x**4 + 2* c**2*x**2 + 1))