\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^2} \, dx\) [246]

Optimal result

Integrand size = 26, antiderivative size = 279 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {2 b^2 x}{c^4 d^2}+\frac {b (a+b \text {arcsinh}(c x))}{c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}-\frac {b^2 \arctan (c x)}{c^5 d^2}+\frac {3 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}-\frac {3 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^5 d^2} \] Output:

2*b^2*x/c^4/d^2+b*(a+b*arcsinh(c*x))/c^5/d^2/(c^2*x^2+1)^(1/2)-2*b*(c^2*x^ 
2+1)^(1/2)*(a+b*arcsinh(c*x))/c^5/d^2+3/2*x*(a+b*arcsinh(c*x))^2/c^4/d^2-1 
/2*x^3*(a+b*arcsinh(c*x))^2/c^2/d^2/(c^2*x^2+1)-3*(a+b*arcsinh(c*x))^2*arc 
tan(c*x+(c^2*x^2+1)^(1/2))/c^5/d^2-b^2*arctan(c*x)/c^5/d^2+3*I*b*(a+b*arcs 
inh(c*x))*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^5/d^2-3*I*b*(a+b*arcsinh 
(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c^5/d^2-3*I*b^2*polylog(3,-I*( 
c*x+(c^2*x^2+1)^(1/2)))/c^5/d^2+3*I*b^2*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2) 
))/c^5/d^2
 

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.73 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\frac {2 a^2 x}{c^4}+\frac {a^2 x}{c^4+c^6 x^2}-\frac {3 a^2 \arctan (c x)}{c^5}-\frac {2 a b \left (\sqrt {1+c^2 x^2}+2 c^2 x^2 \sqrt {1+c^2 x^2}-3 c x \text {arcsinh}(c x)-2 c^3 x^3 \text {arcsinh}(c x)+3 i \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+3 i c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-3 i \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-3 i c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-3 i \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+3 i \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{c^5+c^7 x^2}+\frac {2 b^2 \left (\frac {\text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\frac {c x \text {arcsinh}(c x)^2}{2+2 c^2 x^2}+c x \left (2+\text {arcsinh}(c x)^2\right )+\frac {1}{2} i \left (4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+3 \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-3 \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+6 \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-6 \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )\right )}{c^5}}{2 d^2} \] Input:

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

Output:

((2*a^2*x)/c^4 + (a^2*x)/(c^4 + c^6*x^2) - (3*a^2*ArcTan[c*x])/c^5 - (2*a* 
b*(Sqrt[1 + c^2*x^2] + 2*c^2*x^2*Sqrt[1 + c^2*x^2] - 3*c*x*ArcSinh[c*x] - 
2*c^3*x^3*ArcSinh[c*x] + (3*I)*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (3 
*I)*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (3*I)*ArcSinh[c*x]*Lo 
g[1 + I*E^ArcSinh[c*x]] - (3*I)*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c 
*x]] - (3*I)*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (3*I)*(1 + c^ 
2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]]))/(c^5 + c^7*x^2) + (2*b^2*(ArcSinh[c* 
x]/Sqrt[1 + c^2*x^2] - 2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + (c*x*ArcSinh[c*x 
]^2)/(2 + 2*c^2*x^2) + c*x*(2 + ArcSinh[c*x]^2) + (I/2)*((4*I)*ArcTan[Tanh 
[ArcSinh[c*x]/2]] + 3*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - 3*ArcSinh 
[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 6*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSi 
nh[c*x]] - 6*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 6*PolyLog[3, (-I) 
/E^ArcSinh[c*x]] - 6*PolyLog[3, I/E^ArcSinh[c*x]])))/c^5)/(2*d^2)
 

Rubi [A] (verified)

Time = 2.92 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {6225, 27, 6219, 27, 299, 216, 6227, 6204, 3042, 4668, 3011, 2720, 6213, 24, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6219

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{2 c^2 d^2}+\frac {b \left (-b c \int \frac {c^2 x^2+2}{c^4 \left (c^2 x^2+1\right )}dx+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{2 c^2 d^2}+\frac {b \left (-\frac {b \int \frac {c^2 x^2+2}{c^2 x^2+1}dx}{c^3}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{2 c^2 d^2}+\frac {b \left (-\frac {b \left (\int \frac {1}{c^2 x^2+1}dx+x\right )}{c^3}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {3 \left (-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{c^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6204

\(\displaystyle \frac {3 \left (-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {3 \left (-\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {3 \left (-\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {3 \left (-\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {3 \left (-\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {3 \left (-\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {3 \left (-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{c^3}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{c}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2}\right )}{2 c^2 d^2}+\frac {b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^4}+\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{c^3}\right )}{c d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

Input:

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

Output:

-1/2*(x^3*(a + b*ArcSinh[c*x])^2)/(c^2*d^2*(1 + c^2*x^2)) + (b*((a + b*Arc 
Sinh[c*x])/(c^4*Sqrt[1 + c^2*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x 
]))/c^4 - (b*(x + ArcTan[c*x]/c))/c^3))/(c*d^2) + (3*((x*(a + b*ArcSinh[c* 
x])^2)/c^2 - (2*b*(-((b*x)/c) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c 
^2))/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^A 
rcSinh[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]]))/c^3))/(2*c^2*d^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 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 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 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6219

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) 
/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

rule 6225

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^( 
m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a**2*x**4/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(b**2*x**4 
*asinh(c*x)**2/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(2*a*b*x**4*asi 
nh(c*x)/(c**4*x**4 + 2*c**2*x**2 + 1), x))/d**2
 

Maxima [F]

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

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

Output:

1/2*a^2*(x/(c^6*d^2*x^2 + c^4*d^2) + 2*x/(c^4*d^2) - 3*arctan(c*x)/(c^5*d^ 
2)) + integrate(b^2*x^4*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^4*d^2*x^4 + 2*c^ 
2*d^2*x^2 + d^2) + 2*a*b*x^4*log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^4 + 2 
*c^2*d^2*x^2 + d^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {-3 \mathit {atan} \left (c x \right ) a^{2} c^{2} x^{2}-3 \mathit {atan} \left (c x \right ) a^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{c^{4} x^{4}+2 c^{2} x^{2}+1}d x \right ) a b \,c^{7} x^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{c^{4} x^{4}+2 c^{2} x^{2}+1}d x \right ) a b \,c^{5}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{c^{4} x^{4}+2 c^{2} x^{2}+1}d x \right ) b^{2} c^{7} x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{c^{4} x^{4}+2 c^{2} x^{2}+1}d x \right ) b^{2} c^{5}+2 a^{2} c^{3} x^{3}+3 a^{2} c x}{2 c^{5} d^{2} \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 3*atan(c*x)*a**2*c**2*x**2 - 3*atan(c*x)*a**2 + 4*int((asinh(c*x)*x**4 
)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*a*b*c**7*x**2 + 4*int((asinh(c*x)*x**4) 
/(c**4*x**4 + 2*c**2*x**2 + 1),x)*a*b*c**5 + 2*int((asinh(c*x)**2*x**4)/(c 
**4*x**4 + 2*c**2*x**2 + 1),x)*b**2*c**7*x**2 + 2*int((asinh(c*x)**2*x**4) 
/(c**4*x**4 + 2*c**2*x**2 + 1),x)*b**2*c**5 + 2*a**2*c**3*x**3 + 3*a**2*c* 
x)/(2*c**5*d**2*(c**2*x**2 + 1))