\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 (d+c^2 d x^2)} \, dx\) [245]

Optimal result

Integrand size = 26, antiderivative size = 297 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {14 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \] Output:

-1/3*b^2*c^2/d/x-1/3*b*c*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/d/x^2-1/3*(a 
+b*arcsinh(c*x))^2/d/x^3+c^2*(a+b*arcsinh(c*x))^2/d/x+2*c^3*(a+b*arcsinh(c 
*x))^2*arctan(c*x+(c^2*x^2+1)^(1/2))/d+14/3*b*c^3*(a+b*arcsinh(c*x))*arcta 
nh(c*x+(c^2*x^2+1)^(1/2))/d+7/3*b^2*c^3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/ 
d-2*I*b*c^3*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d+2*I 
*b*c^3*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d-7/3*b^2*c 
^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d+2*I*b^2*c^3*polylog(3,-I*(c*x+(c^2*x 
^2+1)^(1/2)))/d-2*I*b^2*c^3*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2)))/d
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(297)=594\).

Time = 7.24 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)),x]
 

Output:

-1/3*a^2/(d*x^3) + (a^2*c^2)/(d*x) + (a^2*c^3*ArcTan[c*x])/d + (2*a*b*(-1/ 
6*(c*Sqrt[1 + c^2*x^2])/x^2 - ArcSinh[c*x]/(3*x^3) + (c^3*ArcTanh[Sqrt[1 + 
 c^2*x^2]])/6 - c^2*(-(ArcSinh[c*x]/x) - c*ArcTanh[Sqrt[1 + c^2*x^2]]) - ( 
I/2)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x] 
])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c) + (I/2)*c^4*(-1/2*ArcSinh[c* 
x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, I*E^ 
ArcSinh[c*x]])/c)))/d + (b^2*c^3*(-4*Coth[ArcSinh[c*x]/2] + 14*ArcSinh[c*x 
]^2*Coth[ArcSinh[c*x]/2] - 2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - (c*x*Ar 
cSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 56*ArcSinh[c*x]*Log[1 - E^(-ArcSi 
nh[c*x])] - (24*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSi 
nh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 56*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[ 
c*x])] - 56*PolyLog[2, -E^(-ArcSinh[c*x])] - (48*I)*ArcSinh[c*x]*PolyLog[2 
, (-I)/E^ArcSinh[c*x]] + (48*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] 
+ 56*PolyLog[2, E^(-ArcSinh[c*x])] - (48*I)*PolyLog[3, (-I)/E^ArcSinh[c*x] 
] + (48*I)*PolyLog[3, I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sech[ArcSinh[c*x] 
/2]^2 - (8*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2]^4)/(c^3*x^3) + 4*Tanh[ArcSi 
nh[c*x]/2] - 14*ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d)
 

Rubi [A] (verified)

Time = 2.58 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {6224, 27, 6224, 15, 6204, 3042, 4668, 3011, 2720, 6231, 3042, 26, 4670, 2715, 2838, 7143}

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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (c^2 d x^2+d\right )} \, dx\)

\(\Big \downarrow \) 6224

\(\displaystyle c^2 \left (-\int \frac {(a+b \text {arcsinh}(c x))^2}{d x^2 \left (c^2 x^2+1\right )}dx\right )+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {c^2 x^2+1}}dx}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 x^2+1\right )}dx}{d}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {c^2 x^2+1}}dx}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6224

\(\displaystyle -\frac {c^2 \left (c^2 \left (-\int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {1}{2} b c \int \frac {1}{x^2}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {c^2 \left (c^2 \left (-\int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6204

\(\displaystyle -\frac {c^2 \left (2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-c \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {c^2 \left (2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-c \int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {c^2 \left (-c \left (-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6231

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} c^2 \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 i b c \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} i c^2 \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 i b c \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} i c^2 \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 i b c \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} i c^2 \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {c^2 \left (-c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+2 i b c \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} i c^2 \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {c^2 \left (-c \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )+2 i b c \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x}\right )}{d}+\frac {2 b c \left (-\frac {1}{2} i c^2 \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

Input:

Int[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)),x]
 

Output:

-1/3*(a + b*ArcSinh[c*x])^2/(d*x^3) + (2*b*c*(-1/2*(b*c)/x - (Sqrt[1 + c^2 
*x^2]*(a + b*ArcSinh[c*x]))/(2*x^2) - (I/2)*c^2*((2*I)*(a + b*ArcSinh[c*x] 
)*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[ 
2, E^ArcSinh[c*x]])))/(3*d) - (c^2*(-((a + b*ArcSinh[c*x])^2/x) + (2*I)*b* 
c*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ 
ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]) - c*(2*(a + b*ArcSinh[c*x] 
)^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, (- 
I)*E^ArcSinh[c*x]]) + b*PolyLog[3, (-I)*E^ArcSinh[c*x]]) - (2*I)*b*(-((a + 
 b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]]) + b*PolyLog[3, I*E^ArcSinh[ 
c*x]]))))/d
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2715

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]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

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]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4668

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]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6204

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]
 

rule 6224

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]
 

rule 6231

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{x^{4} \left (c^{2} d \,x^{2}+d \right )}d x\]

Input:

int((a+b*arcsinh(x*c))^2/x^4/(c^2*d*x^2+d),x)
 

Output:

int((a+b*arcsinh(x*c))^2/x^4/(c^2*d*x^2+d),x)
 

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d),x, algorithm="fricas")
 

Output:

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^2*d*x^6 + d*x^ 
4), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx}{d} \] Input:

integrate((a+b*asinh(c*x))**2/x**4/(c**2*d*x**2+d),x)
 

Output:

(Integral(a**2/(c**2*x**6 + x**4), x) + Integral(b**2*asinh(c*x)**2/(c**2* 
x**6 + x**4), x) + Integral(2*a*b*asinh(c*x)/(c**2*x**6 + x**4), x))/d
 

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d),x, algorithm="maxima")
 

Output:

1/3*(3*c^3*arctan(c*x)/d + (3*c^2*x^2 - 1)/(d*x^3))*a^2 + integrate(b^2*lo 
g(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^6 + d*x^4) + 2*a*b*log(c*x + sqrt(c^ 
2*x^2 + 1))/(c^2*d*x^6 + d*x^4), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,\left (d\,c^2\,x^2+d\right )} \,d x \] Input:

int((a + b*asinh(c*x))^2/(x^4*(d + c^2*d*x^2)),x)
 

Output:

int((a + b*asinh(c*x))^2/(x^4*(d + c^2*d*x^2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\frac {3 \mathit {atan} \left (c x \right ) a^{2} c^{3} x^{3}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{2} x^{6}+x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{2} x^{6}+x^{4}}d x \right ) b^{2} x^{3}+3 a^{2} c^{2} x^{2}-a^{2}}{3 d \,x^{3}} \] Input:

int((a+b*asinh(c*x))^2/x^4/(c^2*d*x^2+d),x)
 

Output:

(3*atan(c*x)*a**2*c**3*x**3 + 6*int(asinh(c*x)/(c**2*x**6 + x**4),x)*a*b*x 
**3 + 3*int(asinh(c*x)**2/(c**2*x**6 + x**4),x)*b**2*x**3 + 3*a**2*c**2*x* 
*2 - a**2)/(3*d*x**3)