\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 (\pi +c^2 \pi x^2)^{5/2}} \, dx\) [119]

Optimal result

Integrand size = 26, antiderivative size = 222 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b c}{2 \pi ^{5/2} x}+\frac {b c^3 x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}+\frac {5 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {5 b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}}-\frac {5 b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}} \] Output:

-1/2*b*c/Pi^(5/2)/x+1/6*b*c^3*x/Pi^(5/2)/(c^2*x^2+1)-5/6*c^2*(a+b*arcsinh( 
c*x))/Pi/(Pi*c^2*x^2+Pi)^(3/2)-1/2*(a+b*arcsinh(c*x))/Pi/x^2/(Pi*c^2*x^2+P 
i)^(3/2)-5/2*c^2*(a+b*arcsinh(c*x))/Pi^2/(Pi*c^2*x^2+Pi)^(1/2)+13/6*b*c^2* 
arctan(c*x)/Pi^(5/2)+5*c^2*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2 
))/Pi^(5/2)+5/2*b*c^2*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/Pi^(5/2)-5/2*b*c^2 
*polylog(2,c*x+(c^2*x^2+1)^(1/2))/Pi^(5/2)
 

Mathematica [A] (verified)

Time = 3.88 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.49 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-\frac {8 a c^2}{\left (1+c^2 x^2\right )^{3/2}}+\frac {4 b c^3 x}{1+c^2 x^2}-\frac {48 a c^2}{\sqrt {1+c^2 x^2}}-\frac {12 a \sqrt {1+c^2 x^2}}{x^2}-\frac {56 b c^2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}-\frac {48 b c^4 x^2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+104 b c^2 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-6 b c^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-3 b c^2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-60 b c^2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+60 b c^2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-60 a c^2 \log (x)+60 a c^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )-60 b c^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+60 b c^2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+6 b c^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\frac {6 b c \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{x}}{24 \pi ^{5/2}} \] Input:

Integrate[(a + b*ArcSinh[c*x])/(x^3*(Pi + c^2*Pi*x^2)^(5/2)),x]
 

Output:

((-8*a*c^2)/(1 + c^2*x^2)^(3/2) + (4*b*c^3*x)/(1 + c^2*x^2) - (48*a*c^2)/S 
qrt[1 + c^2*x^2] - (12*a*Sqrt[1 + c^2*x^2])/x^2 - (56*b*c^2*ArcSinh[c*x])/ 
(1 + c^2*x^2)^(3/2) - (48*b*c^4*x^2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + 10 
4*b*c^2*ArcTan[Tanh[ArcSinh[c*x]/2]] - 6*b*c^2*Coth[ArcSinh[c*x]/2] - 3*b* 
c^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 60*b*c^2*ArcSinh[c*x]*Log[1 - E^ 
(-ArcSinh[c*x])] + 60*b*c^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 60*a 
*c^2*Log[x] + 60*a*c^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] - 60*b*c^2*PolyLog[ 
2, -E^(-ArcSinh[c*x])] + 60*b*c^2*PolyLog[2, E^(-ArcSinh[c*x])] + 6*b*c^2* 
Tanh[ArcSinh[c*x]/2] - (6*b*c*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2])/x)/(24*Pi 
^(5/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.40 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {6224, 253, 264, 216, 6226, 215, 216, 6226, 216, 6231, 3042, 26, 4670, 2715, 2838}

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.

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi c^2 x^2+\pi \right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 253

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcSinh[c*x])/(x^3*(Pi + c^2*Pi*x^2)^(5/2)),x]
 

Output:

-1/2*(a + b*ArcSinh[c*x])/(Pi*x^2*(Pi + c^2*Pi*x^2)^(3/2)) + (b*c*(1/(2*x* 
(1 + c^2*x^2)) + (3*(-x^(-1) - c*ArcTan[c*x]))/2))/(2*Pi^(5/2)) - (5*c^2*( 
(a + b*ArcSinh[c*x])/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (b*c*(x/(2*(1 + c^2* 
x^2)) + ArcTan[c*x]/(2*c)))/(3*Pi^(5/2)) + ((a + b*ArcSinh[c*x])/(Pi*Sqrt[ 
Pi + c^2*Pi*x^2]) - (b*ArcTan[c*x])/Pi^(3/2) + (I*((2*I)*(a + b*ArcSinh[c* 
x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLo 
g[2, E^ArcSinh[c*x]]))/Pi^(3/2))/Pi))/2
 

Defintions of rubi rules used

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 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 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, 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 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 3042

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

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 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 6226

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
 + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ 
b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 
1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

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]
 

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.41

method	result	size
default	\(a \left (-\frac {1}{2 \pi \,x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {5 c^{2} \left (\frac {1}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}}{\pi }\right )}{2}\right )-\frac {5 b \,x^{2} \operatorname {arcsinh}\left (x c \right ) c^{4}}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b \,c^{3} x}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )}-\frac {10 b \,\operatorname {arcsinh}\left (x c \right ) c^{2}}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b c}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right ) x}-\frac {b \,\operatorname {arcsinh}\left (x c \right )}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}+\frac {13 b \,c^{2} \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}\)	\(314\)
parts	\(a \left (-\frac {1}{2 \pi \,x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {5 c^{2} \left (\frac {1}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}}{\pi }\right )}{2}\right )-\frac {5 b \,x^{2} \operatorname {arcsinh}\left (x c \right ) c^{4}}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b \,c^{3} x}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )}-\frac {10 b \,\operatorname {arcsinh}\left (x c \right ) c^{2}}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b c}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right ) x}-\frac {b \,\operatorname {arcsinh}\left (x c \right )}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}+\frac {13 b \,c^{2} \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}\)	\(314\)

Input:

int((a+b*arcsinh(x*c))/x^3/(Pi*c^2*x^2+Pi)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

a*(-1/2/Pi/x^2/(Pi*c^2*x^2+Pi)^(3/2)-5/2*c^2*(1/3/Pi/(Pi*c^2*x^2+Pi)^(3/2) 
+1/Pi*(1/Pi/(Pi*c^2*x^2+Pi)^(1/2)-1/Pi^(3/2)*arctanh(Pi^(1/2)/(Pi*c^2*x^2+ 
Pi)^(1/2)))))-5/2*b/Pi^(5/2)/(c^2*x^2+1)^(3/2)*x^2*arcsinh(x*c)*c^4-1/3*b* 
c^3*x/Pi^(5/2)/(c^2*x^2+1)-10/3*b/Pi^(5/2)/(c^2*x^2+1)^(3/2)*arcsinh(x*c)* 
c^2-1/2*b/Pi^(5/2)/(c^2*x^2+1)/x*c-1/2*b/Pi^(5/2)/(c^2*x^2+1)^(3/2)/x^2*ar 
csinh(x*c)+13/3*b/Pi^(5/2)*c^2*arctan(x*c+(c^2*x^2+1)^(1/2))+5/2*b/Pi^(5/2 
)*c^2*dilog(1+x*c+(c^2*x^2+1)^(1/2))+5/2*b/Pi^(5/2)*c^2*arcsinh(x*c)*ln(1+ 
x*c+(c^2*x^2+1)^(1/2))+5/2*b/Pi^(5/2)*c^2*dilog(x*c+(c^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(pi*c^2*x^2+pi)^(5/2),x, algorithm="frica 
s")
 

Output:

integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^3*c^6*x^9 + 3*pi^3 
*c^4*x^7 + 3*pi^3*c^2*x^5 + pi^3*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a}{c^{4} x^{7} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} + x^{3} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{7} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} + x^{3} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \] Input:

integrate((a+b*asinh(c*x))/x**3/(pi*c**2*x**2+pi)**(5/2),x)
 

Output:

(Integral(a/(c**4*x**7*sqrt(c**2*x**2 + 1) + 2*c**2*x**5*sqrt(c**2*x**2 + 
1) + x**3*sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(c**4*x**7*sqrt 
(c**2*x**2 + 1) + 2*c**2*x**5*sqrt(c**2*x**2 + 1) + x**3*sqrt(c**2*x**2 + 
1)), x))/pi**(5/2)
 

Maxima [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(pi*c^2*x^2+pi)^(5/2),x, algorithm="maxim 
a")
 

Output:

1/6*a*(15*c^2*arcsinh(1/(c*abs(x)))/pi^(5/2) - 5*c^2/(pi*(pi + pi*c^2*x^2) 
^(3/2)) - 15*c^2/(pi^2*sqrt(pi + pi*c^2*x^2)) - 3/(pi*(pi + pi*c^2*x^2)^(3 
/2)*x^2)) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/((pi + pi*c^2*x^2)^(5 
/2)*x^3), x)
 

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(pi*c^2*x^2+pi)^(5/2),x, algorithm="giac" 
)
 

Output:

integrate((b*arcsinh(c*x) + a)/((pi + pi*c^2*x^2)^(5/2)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \] Input:

int((a + b*asinh(c*x))/(x^3*(Pi + Pi*c^2*x^2)^(5/2)),x)
 

Output:

int((a + b*asinh(c*x))/(x^3*(Pi + Pi*c^2*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-15 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a +6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b \,c^{4} x^{6}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b \,c^{2} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{7}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b \,x^{2}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a \,c^{6} x^{6}-30 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a \,c^{4} x^{4}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a \,c^{2} x^{2}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \,c^{6} x^{6}+30 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \,c^{4} x^{4}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \,c^{2} x^{2}}{6 \sqrt {\pi }\, \pi ^{2} x^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))/x^3/(Pi*c^2*x^2+Pi)^(5/2),x)
 

Output:

( - 15*sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 20*sqrt(c**2*x**2 + 1)*a*c**2*x** 
2 - 3*sqrt(c**2*x**2 + 1)*a + 6*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x 
**7 + 2*sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3),x)*b*c** 
4*x**6 + 12*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**7 + 2*sqrt(c**2*x* 
*2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3),x)*b*c**2*x**4 + 6*int(asinh 
(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**7 + 2*sqrt(c**2*x**2 + 1)*c**2*x**5 + s 
qrt(c**2*x**2 + 1)*x**3),x)*b*x**2 - 15*log(sqrt(c**2*x**2 + 1) + c*x - 1) 
*a*c**6*x**6 - 30*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a*c**4*x**4 - 15*log( 
sqrt(c**2*x**2 + 1) + c*x - 1)*a*c**2*x**2 + 15*log(sqrt(c**2*x**2 + 1) + 
c*x + 1)*a*c**6*x**6 + 30*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a*c**4*x**4 + 
 15*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a*c**2*x**2)/(6*sqrt(pi)*pi**2*x**2 
*(c**4*x**4 + 2*c**2*x**2 + 1))