3.1.45 \(\int \frac {a+b \arcsin (c x)}{x^4 (d-c^2 d x^2)^2} \, dx\) [45]

3.1.45.1 Optimal result

Integrand size = 25, antiderivative size = 259 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c^3}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{2 d^2}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 d^2} \]

output

1/3*(-a-b*arcsin(c*x))/d^2/x^3/(-c^2*x^2+1)-5/3*c^2*(a+b*arcsin(c*x))/d^2/ 
x/(-c^2*x^2+1)+5/2*c^4*x*(a+b*arcsin(c*x))/d^2/(-c^2*x^2+1)-5*I*c^3*(a+b*a 
rcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^2-13/6*b*c^3*arctanh((-c^2* 
x^2+1)^(1/2))/d^2+5/2*I*b*c^3*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2 
-5/2*I*b*c^3*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2-1/3*b*c^3/d^2/(-c 
^2*x^2+1)^(1/2)-1/6*b*c/d^2/x^2/(-c^2*x^2+1)^(1/2)
 

3.1.45.2 Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.64 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {\frac {4 a}{x^3}+\frac {24 a c^2}{x}+\frac {2 b c \sqrt {1-c^2 x^2}}{x^2}-\frac {3 b c^3 \sqrt {1-c^2 x^2}}{-1+c x}+\frac {3 b c^3 \sqrt {1-c^2 x^2}}{1+c x}+\frac {6 a c^4 x}{-1+c^2 x^2}+15 i b c^3 \pi \arcsin (c x)+\frac {4 b \arcsin (c x)}{x^3}+\frac {24 b c^2 \arcsin (c x)}{x}+\frac {3 b c^3 \arcsin (c x)}{-1+c x}+\frac {3 b c^3 \arcsin (c x)}{1+c x}+26 b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-15 b c^3 \pi \log \left (1-i e^{i \arcsin (c x)}\right )-30 b c^3 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-15 b c^3 \pi \log \left (1+i e^{i \arcsin (c x)}\right )+30 b c^3 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+15 a c^3 \log (1-c x)-15 a c^3 \log (1+c x)+15 b c^3 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+15 b c^3 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-30 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+30 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{12 d^2} \]

input

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

output

-1/12*((4*a)/x^3 + (24*a*c^2)/x + (2*b*c*Sqrt[1 - c^2*x^2])/x^2 - (3*b*c^3 
*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (3*b*c^3*Sqrt[1 - c^2*x^2])/(1 + c*x) + ( 
6*a*c^4*x)/(-1 + c^2*x^2) + (15*I)*b*c^3*Pi*ArcSin[c*x] + (4*b*ArcSin[c*x] 
)/x^3 + (24*b*c^2*ArcSin[c*x])/x + (3*b*c^3*ArcSin[c*x])/(-1 + c*x) + (3*b 
*c^3*ArcSin[c*x])/(1 + c*x) + 26*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 15*b*c 
^3*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 30*b*c^3*ArcSin[c*x]*Log[1 - I*E^(I*A 
rcSin[c*x])] - 15*b*c^3*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 30*b*c^3*ArcSin[ 
c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 15*a*c^3*Log[1 - c*x] - 15*a*c^3*Log[1 
 + c*x] + 15*b*c^3*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 15*b*c^3*Pi*Log[ 
Sin[(Pi + 2*ArcSin[c*x])/4]] - (30*I)*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c* 
x])] + (30*I)*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2
 

3.1.45.3 Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.17, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {5204, 27, 243, 52, 61, 73, 221, 5204, 243, 61, 73, 221, 5162, 241, 5164, 3042, 4669, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {5 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}+\frac {b c \int \frac {1}{x^4 \left (1-c^2 x^2\right )^{3/2}}dx^2}{6 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {5 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}+\frac {b c \left (\frac {3}{2} c^2 \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx^2-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {5 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}+\frac {b c \left (\frac {3}{2} c^2 \left (\int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {2}{\sqrt {1-c^2 x^2}}\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-\frac {2 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2}\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 5204

\(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+b c \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2}}dx-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx^2-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {2}{\sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-\frac {2 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 5162

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 5164

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 4669

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\right )}{3 d^2}-\frac {a+b \arcsin (c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {3}{2} c^2 \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {1}{x^2 \sqrt {1-c^2 x^2}}\right )}{6 d^2}\)

input

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

output

-1/3*(a + b*ArcSin[c*x])/(d^2*x^3*(1 - c^2*x^2)) + (b*c*(-(1/(x^2*Sqrt[1 - 
 c^2*x^2])) + (3*c^2*(2/Sqrt[1 - c^2*x^2] - 2*ArcTanh[Sqrt[1 - c^2*x^2]])) 
/2))/(6*d^2) + (5*c^2*(-((a + b*ArcSin[c*x])/(x*(1 - c^2*x^2))) + (b*c*(2/ 
Sqrt[1 - c^2*x^2] - 2*ArcTanh[Sqrt[1 - c^2*x^2]]))/2 + 3*c^2*(-1/2*b/(c*Sq 
rt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(2*(1 - c^2*x^2)) + ((-2*I)*(a 
+ b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, (-I)*E^(I*ArcS 
in[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*c))))/(3*d^2)
 

3.1.45.3.1 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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 241

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5162

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[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*Ar 
cSin[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*ArcSin[c*x])^(n - 1), x], x 
]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] 
/; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5204

Int[((a_.) + ArcSin[(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 
*ArcSin[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*ArcSin[c*x])^n, x], x] - Simp[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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

3.1.45.4 Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(312\)
default	\(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(312\)
parts	\(\frac {a \left (-\frac {1}{3 x^{3}}-\frac {2 c^{2}}{x}-\frac {c^{3}}{4 \left (c x -1\right )}-\frac {5 c^{3} \ln \left (c x -1\right )}{4}-\frac {c^{3}}{4 \left (c x +1\right )}+\frac {5 c^{3} \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \,c^{3} \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\)	\(320\)

input

int((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
 

output

c^3*(a/d^2*(-1/3/c^3/x^3-2/c/x-1/4/(c*x-1)-5/4*ln(c*x-1)-1/4/(c*x+1)+5/4*l 
n(c*x+1))+b/d^2*(-1/6*(15*c^4*x^4*arcsin(c*x)-2*c^3*x^3*(-c^2*x^2+1)^(1/2) 
-10*c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(c*x))/(c^2*x^2-1)/ 
c^3/x^3+13/6*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-13/6*ln(1+I*c*x+(-c^2*x^2+1)^( 
1/2))-5/2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5/2*arcsin(c*x)*l 
n(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5/2*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2 
)))-5/2*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
 

3.1.45.5 Fricas [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]

input

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

output

integral((b*arcsin(c*x) + a)/(c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4), x)
 

3.1.45.6 Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \]

input

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

output

(Integral(a/(c**4*x**8 - 2*c**2*x**6 + x**4), x) + Integral(b*asin(c*x)/(c 
**4*x**8 - 2*c**2*x**6 + x**4), x))/d**2
 

3.1.45.7 Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]

input

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

output

1/12*(15*c^3*log(c*x + 1)/d^2 - 15*c^3*log(c*x - 1)/d^2 - 2*(15*c^4*x^4 - 
10*c^2*x^2 - 2)/(c^2*d^2*x^5 - d^2*x^3))*a + 1/12*(15*(c^5*x^5 - c^3*x^3)* 
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 15*(c^5*x^5 - c^ 
3*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(15*c^ 
4*x^4 - 10*c^2*x^2 - 2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 12*(c 
^2*d^2*x^5 - d^2*x^3)*integrate(-1/12*(30*c^5*x^4 - 20*c^3*x^2 - 15*(c^6*x 
^5 - c^4*x^3)*log(c*x + 1) + 15*(c^6*x^5 - c^4*x^3)*log(-c*x + 1) - 4*c)*s 
qrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x))*b 
/(c^2*d^2*x^5 - d^2*x^3)
 

3.1.45.8 Giac [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]

input

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

output

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

3.1.45.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

input

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

output

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