Integrand size = 28, antiderivative size = 687 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x (a+b \text {arcsinh}(c x))}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \]
-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)+1/3*b^2*c^2/d^2/(c^2*d*x^2+d)^(1/2)-5/2*c^2*( a+b*arcsinh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)-b*c*(a+b*arcsinh(c*x))/d^2/x/( c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-2/3*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)+26/3*b*c^2*(a+b*arcsinh(c*x))*arctan(c *x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+5*c^2*(a+b *arcsinh(c*x))^2*arctanh(c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2 *d*x^2+d)^(1/2)-b^2*c^2*arctanh((c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/( c^2*d*x^2+d)^(1/2)+5*b*c^2*(a+b*arcsinh(c*x))*polylog(2,-c*x-(c^2*x^2+1)^( 1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-13/3*I*b^2*c^2*polylog(2,- I*(c*x+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+13/3* I*b^2*c^2*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/2)/d^2/(c^2* d*x^2+d)^(1/2)-5*b*c^2*(a+b*arcsinh(c*x))*polylog(2,c*x+(c^2*x^2+1)^(1/2)) *(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-5*b^2*c^2*polylog(3,-c*x-(c^2*x ^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+5*b^2*c^2*polylog(3 ,c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)
Time = 7.78 (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=\sqrt {d \left (1+c^2 x^2\right )} \left (-\frac {a^2}{2 d^3 x^2}-\frac {a^2 c^2}{3 d^3 \left (1+c^2 x^2\right )^2}-\frac {2 a^2 c^2}{d^3 \left (1+c^2 x^2\right )}\right )-\frac {5 a^2 c^2 \log (x)}{2 d^{5/2}}+\frac {5 a^2 c^2 \log \left (d+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )}{2 d^{5/2}}+\frac {a b c^2 \left (\frac {4 c x}{\sqrt {1+c^2 x^2}}-48 \text {arcsinh}(c x)-\frac {8 \text {arcsinh}(c x)}{1+c^2 x^2}+104 \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-6 \sqrt {1+c^2 x^2} \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-60 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+60 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-60 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+60 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+6 \sqrt {1+c^2 x^2} \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{12 d^2 \sqrt {d \left (1+c^2 x^2\right )}}+\frac {b^2 c^2 \left (8+\frac {8 c x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-48 \text {arcsinh}(c x)^2-\frac {8 \text {arcsinh}(c x)^2}{1+c^2 x^2}-12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-60 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )-104 i \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+104 i \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+60 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+24 \sqrt {1+c^2 x^2} \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-120 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-104 i \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+104 i \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+120 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-120 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )+120 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )-3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{24 d^2 \sqrt {d \left (1+c^2 x^2\right )}} \]
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 |
| \(\displaystyle -\frac {5}{2} c^2 \left (\frac {\frac {i \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{d \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}}{d}-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {b c \sqrt {c^2 x^2+1} \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}\) |
3.4.18.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x ], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ b*c*(n/(2*f*(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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]