\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [255]

Optimal result

Integrand size = 26, antiderivative size = 320 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {b^2 x}{12 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {b (a+b \text {arcsinh}(c x))}{6 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b (a+b \text {arcsinh}(c x))}{4 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))^2}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}+\frac {7 b^2 \arctan (c x)}{6 c^5 d^3}-\frac {3 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}+\frac {3 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3} \] Output:

-1/12*b^2*x/c^4/d^3/(c^2*x^2+1)+1/6*b*(a+b*arcsinh(c*x))/c^5/d^3/(c^2*x^2+ 
1)^(3/2)-5/4*b*(a+b*arcsinh(c*x))/c^5/d^3/(c^2*x^2+1)^(1/2)-1/4*x^3*(a+b*a 
rcsinh(c*x))^2/c^2/d^3/(c^2*x^2+1)^2-3/8*x*(a+b*arcsinh(c*x))^2/c^4/d^3/(c 
^2*x^2+1)+3/4*(a+b*arcsinh(c*x))^2*arctan(c*x+(c^2*x^2+1)^(1/2))/c^5/d^3+7 
/6*b^2*arctan(c*x)/c^5/d^3-3/4*I*b*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c 
^2*x^2+1)^(1/2)))/c^5/d^3+3/4*I*b*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2 
*x^2+1)^(1/2)))/c^5/d^3+3/4*I*b^2*polylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^ 
5/d^3-3/4*I*b^2*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2)))/c^5/d^3
 

Mathematica [A] (verified)

Time = 2.23 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.72 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\frac {6 a^2 c x}{\left (1+c^2 x^2\right )^2}-\frac {15 a^2 c x}{1+c^2 x^2}+\frac {15 a b \left (\sqrt {1+c^2 x^2}-i \text {arcsinh}(c x)\right )}{-1+i c x}+\frac {15 a b \left (\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)\right )}{-1-i c x}-\frac {i a b \left ((-2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(-i+c x)^2}+\frac {i a b \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(i+c x)^2}+9 a^2 \arctan (c x)+\frac {9}{2} i a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {9}{2} i a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+b^2 \left (-\frac {2 c x}{1+c^2 x^2}+\frac {4 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}-\frac {30 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {6 c x \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^2}-\frac {15 c x \text {arcsinh}(c x)^2}{1+c^2 x^2}+56 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-9 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+9 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-18 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+18 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-18 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+18 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )}{24 c^5 d^3} \] Input:

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]
 

Output:

((6*a^2*c*x)/(1 + c^2*x^2)^2 - (15*a^2*c*x)/(1 + c^2*x^2) + (15*a*b*(Sqrt[ 
1 + c^2*x^2] - I*ArcSinh[c*x]))/(-1 + I*c*x) + (15*a*b*(Sqrt[1 + c^2*x^2] 
+ I*ArcSinh[c*x]))/(-1 - I*c*x) - (I*a*b*((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 
 3*ArcSinh[c*x]))/(-I + c*x)^2 + (I*a*b*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3 
*ArcSinh[c*x]))/(I + c*x)^2 + 9*a^2*ArcTan[c*x] + ((9*I)/2)*a*b*(ArcSinh[c 
*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^Arc 
Sinh[c*x]]) - ((9*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^Ar 
cSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + b^2*((-2*c*x)/(1 + c^2*x^ 
2) + (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - (30*ArcSinh[c*x])/Sqrt[1 + c^2 
*x^2] + (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 - (15*c*x*ArcSinh[c*x]^2)/( 
1 + c^2*x^2) + 56*ArcTan[Tanh[ArcSinh[c*x]/2]] - (9*I)*ArcSinh[c*x]^2*Log[ 
1 - I/E^ArcSinh[c*x]] + (9*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - ( 
18*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (18*I)*ArcSinh[c*x]*P 
olyLog[2, I/E^ArcSinh[c*x]] - (18*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (18 
*I)*PolyLog[3, I/E^ArcSinh[c*x]]))/(24*c^5*d^3)
 

Rubi [A] (verified)

Time = 2.60 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.03, 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, 298, 216, 6225, 6204, 3042, 4668, 3011, 2720, 6213, 216, 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.

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

\(\Big \downarrow \) 6225

\(\displaystyle \frac {b \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx}{2 c d^3}+\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d^2 \left (c^2 x^2+1\right )^2}dx}{4 c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx}{4 c^2 d^3}+\frac {b \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 6219

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx}{4 c^2 d^3}+\frac {b \left (\frac {b \left (\frac {5}{2} \int \frac {1}{c^2 x^2+1}dx-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 6225

\(\displaystyle \frac {3 \left (\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{2 c^2}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 6204

\(\displaystyle \frac {3 \left (\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c^3}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c^3}+\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {3 \left (\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c^3}+\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {3 \left (\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c^3}+\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {3 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c^3}+\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {3 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c^3}+\frac {b \left (\frac {b \int \frac {1}{c^2 x^2+1}dx}{c}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {3 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c^3}+\frac {b \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {3 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{2 c^3}+\frac {b \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )}{c}-\frac {x (a+b \text {arcsinh}(c x))^2}{2 c^2 \left (c^2 x^2+1\right )}\right )}{4 c^2 d^3}+\frac {b \left (-\frac {a+b \text {arcsinh}(c x)}{c^4 \sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 c^4 \left (c^2 x^2+1\right )^{3/2}}+\frac {b \left (\frac {5 \arctan (c x)}{2 c}-\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c^3}\right )}{2 c d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

Input:

Int[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]
 

Output:

-1/4*(x^3*(a + b*ArcSinh[c*x])^2)/(c^2*d^3*(1 + c^2*x^2)^2) + (b*((a + b*A 
rcSinh[c*x])/(3*c^4*(1 + c^2*x^2)^(3/2)) - (a + b*ArcSinh[c*x])/(c^4*Sqrt[ 
1 + c^2*x^2]) + (b*(-1/2*x/(1 + c^2*x^2) + (5*ArcTan[c*x])/(2*c)))/(3*c^3) 
))/(2*c*d^3) + (3*(-1/2*(x*(a + b*ArcSinh[c*x])^2)/(c^2*(1 + c^2*x^2)) + ( 
b*(-((a + b*ArcSinh[c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[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^ArcSi 
nh[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*c^3)))/(4*c^2*d^3)
 

Defintions of rubi rules used

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 298

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

\[\int \frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{\left (c^{2} d \,x^{2}+d \right )^{3}}d x\]

Input:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^3,x)
 

Output:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^3,x)
 

Fricas [F]

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

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,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^6* 
d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
 

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \] Input:

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**3,x)
 

Output:

(Integral(a**2*x**4/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Inte 
gral(b**2*x**4*asinh(c*x)**2/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), 
x) + Integral(2*a*b*x**4*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 
 + 1), x))/d**3
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="maxima")
 

Output:

-1/8*a^2*((5*c^2*x^3 + 3*x)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - 3*ar 
ctan(c*x)/(c^5*d^3)) + integrate(b^2*x^4*log(c*x + sqrt(c^2*x^2 + 1))^2/(c 
^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3) + 2*a*b*x^4*log(c*x + sq 
rt(c^2*x^2 + 1))/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
 

Giac [F]

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

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)^2*x^4/(c^2*d*x^2 + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \mathit {atan} \left (c x \right ) a^{2} c^{4} x^{4}+6 \mathit {atan} \left (c x \right ) a^{2} c^{2} x^{2}+3 \mathit {atan} \left (c x \right ) a^{2}+16 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{9} x^{4}+32 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{7} x^{2}+16 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{5}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{9} x^{4}+16 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{7} x^{2}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{5}-5 a^{2} c^{3} x^{3}-3 a^{2} c x}{8 c^{5} d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

(3*atan(c*x)*a**2*c**4*x**4 + 6*atan(c*x)*a**2*c**2*x**2 + 3*atan(c*x)*a** 
2 + 16*int((asinh(c*x)*x**4)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x 
)*a*b*c**9*x**4 + 32*int((asinh(c*x)*x**4)/(c**6*x**6 + 3*c**4*x**4 + 3*c* 
*2*x**2 + 1),x)*a*b*c**7*x**2 + 16*int((asinh(c*x)*x**4)/(c**6*x**6 + 3*c* 
*4*x**4 + 3*c**2*x**2 + 1),x)*a*b*c**5 + 8*int((asinh(c*x)**2*x**4)/(c**6* 
x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2*c**9*x**4 + 16*int((asinh(c* 
x)**2*x**4)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2*c**7*x**2 
+ 8*int((asinh(c*x)**2*x**4)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x 
)*b**2*c**5 - 5*a**2*c**3*x**3 - 3*a**2*c*x)/(8*c**5*d**3*(c**4*x**4 + 2*c 
**2*x**2 + 1))