\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [259]

Optimal result

Integrand size = 23, antiderivative size = 309 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {b^2 x}{12 d^3 \left (1+c^2 x^2\right )}+\frac {b (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b (a+b \text {arcsinh}(c x))}{4 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {5 b^2 \arctan (c x)}{6 c d^3}-\frac {3 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c d^3}+\frac {3 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c d^3} \] Output:

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

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\frac {6 a^2 x}{\left (1+c^2 x^2\right )^2}+\frac {9 a^2 x}{1+c^2 x^2}+\frac {9 a^2 \arctan (c x)}{c}+\frac {a b \left (\frac {9 \left (-i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{-i+c x}+\frac {9 \left (i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{i+c x}-\frac {i \left ((-2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(-i+c x)^2}+\frac {i \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(i+c x)^2}+\frac {9}{2} i \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {9}{2} i \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )\right )}{c}+\frac {b^2 \left (-\frac {2 c x}{1+c^2 x^2}+\frac {4 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {18 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {6 c x \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {9 c x \text {arcsinh}(c x)^2}{1+c^2 x^2}-40 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-9 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+9 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-18 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+18 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-18 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+18 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )}{c}}{24 d^3} \] Input:

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

Output:

((6*a^2*x)/(1 + c^2*x^2)^2 + (9*a^2*x)/(1 + c^2*x^2) + (9*a^2*ArcTan[c*x]) 
/c + (a*b*((9*((-I)*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(-I + c*x) + (9*(I* 
Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(I + c*x) - (I*((-2*I + c*x)*Sqrt[1 + c 
^2*x^2] + 3*ArcSinh[c*x]))/(-I + c*x)^2 + (I*((2*I + c*x)*Sqrt[1 + c^2*x^2 
] + 3*ArcSinh[c*x]))/(I + c*x)^2 + ((9*I)/2)*(ArcSinh[c*x]*(ArcSinh[c*x] - 
 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((9*I 
)/2)*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLo 
g[2, I*E^ArcSinh[c*x]])))/c + (b^2*((-2*c*x)/(1 + c^2*x^2) + (4*ArcSinh[c* 
x])/(1 + c^2*x^2)^(3/2) + (18*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (6*c*x*Arc 
Sinh[c*x]^2)/(1 + c^2*x^2)^2 + (9*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 40*A 
rcTan[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]*PolyLog[2, I/E^ArcSi 
nh[c*x]] - (18*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (18*I)*PolyLog[3, I/E^ 
ArcSinh[c*x]]))/c)/(24*d^3)
 

Rubi [A] (verified)

Time = 1.86 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6203, 27, 6203, 6204, 3042, 4668, 3011, 2720, 6213, 215, 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.

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(x*(a + b*ArcSinh[c*x])^2)/(4*d^3*(1 + c^2*x^2)^2) - (b*c*(-1/3*(a + b*Arc 
Sinh[c*x])/(c^2*(1 + c^2*x^2)^(3/2)) + (b*(x/(2*(1 + c^2*x^2)) + ArcTan[c* 
x]/(2*c)))/(3*c)))/(2*d^3) + (3*((x*(a + b*ArcSinh[c*x])^2)/(2*(1 + c^2*x^ 
2)) - b*c*(-((a + b*ArcSinh[c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[c*x 
])/c^2) + (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]]))/(2*c)))/(4*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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 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 6203

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]
 

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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/8*a^2*((3*c^2*x^3 + 5*x)/(c^4*d^3*x^4 + 2*c^2*d^3*x^2 + d^3) + 3*arctan( 
c*x)/(c*d^3)) + integrate(b^2*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*log(c*x + sqrt(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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