3.2.98 \(\int \frac {(a+b \arcsin (c x))^2}{x^2 (d-c^2 d x^2)^2} \, dx\) [198]

3.2.98.1 Optimal result

Integrand size = 27, antiderivative size = 324 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \]

output

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

3.2.98.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1161\) vs. \(2(324)=648\).

Time = 10.44 (sec) , antiderivative size = 1161, normalized size of antiderivative = 3.58 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx =\text {Too large to display} \]

input

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

output

-(a^2/(d^2*x)) - (a^2*c^2*x)/(2*d^2*(-1 + c^2*x^2)) - (3*a^2*c*Log[1 - c*x 
])/(4*d^2) + (3*a^2*c*Log[1 + c*x])/(4*d^2) + (2*a*b*c*((Sqrt[1 - c^2*x^2] 
 - ArcSin[c*x])/(4*(-1 + c*x)) - ArcSin[c*x]/(c*x) - (Sqrt[1 - c^2*x^2] + 
ArcSin[c*x])/(4*(1 + c*x)) - ArcTanh[Sqrt[1 - c^2*x^2]] - (3*(((3*I)/2)*Pi 
*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 
Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x] 
)] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] + Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 
 (2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/4 + (3*((I/2)*Pi*ArcSin[c*x] - 
 (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + Pi*Log[1 - I*E 
^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Log[ 
Cos[ArcSin[c*x]/2]] - Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[ 
2, I*E^(I*ArcSin[c*x])]))/4))/d^2 + (b^2*c*(-4*ArcSin[c*x] - 2*ArcSin[c*x] 
^2*Cot[ArcSin[c*x]/2] + 8*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 6*ArcSi 
n[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 6*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*( 
1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] - 6*ArcSin[c*x]^2*Log 
[1 + I*E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*Ar 
cSin[c*x])))/E^((I/2)*ArcSin[c*x])] + 6*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4 
)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 8*ArcSin[c*x]*Log[1 + 
 E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c 
*x]))/(2*E^((I/2)*ArcSin[c*x]))] - 6*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*Ar...
 

3.2.98.3 Rubi [A] (verified)

Time = 2.67 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5204, 27, 5162, 5164, 3042, 4669, 3011, 2720, 5182, 219, 5208, 219, 5218, 3042, 4671, 2715, 2838, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5208

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.2.98.3.1 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 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 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 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 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 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[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 5182

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

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]
 

rule 5208

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/(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*ArcSin[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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b 
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 5218

Int[(((a_.) + ArcSin[(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*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a 
, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

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]
 

3.2.98.4 Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.87

method	result	size
derivativedivides	\(c \left (\frac {a^{2} \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(605\)
default	\(c \left (\frac {a^{2} \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(605\)
parts	\(\frac {a^{2} \left (-\frac {1}{x}-\frac {c}{4 \left (c x -1\right )}-\frac {3 c \ln \left (c x -1\right )}{4}-\frac {c}{4 \left (c x +1\right )}+\frac {3 c \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} c \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b c \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\)	\(606\)

input

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

output

c*(a^2/d^2*(-1/c/x-1/4/(c*x-1)-3/4*ln(c*x-1)-1/4/(c*x+1)+3/4*ln(c*x+1))+b^ 
2/d^2*(-1/2/c/x/(c^2*x^2-1)*(3*c^2*x^2*arcsin(c*x)-2*c*x*(-c^2*x^2+1)^(1/2 
)-2*arcsin(c*x))*arcsin(c*x)-3/2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^ 
(1/2)))+3*I*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*polylog 
(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^ 
2+1)^(1/2)))-3*I*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3*pol 
ylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+2 
*I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*arctan(I*c*x+(-c^2*x^2+1)^(1/2))- 
2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/2*(3*c^2*x^2*a 
rcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(c*x))/c/x/(c^2*x^2-1)-3/2*arcsi 
n(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+3 
/2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-ln(1+I*c*x+(-c^2*x^2+1)^ 
(1/2))+3/2*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/2*I*dilog(1-I*(I*c*x+ 
(-c^2*x^2+1)^(1/2)))))
 

3.2.98.5 Fricas [F]

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

input

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

output

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

3.2.98.6 Sympy [F]

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

input

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

output

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

3.2.98.7 Maxima [F]

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

input

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

output

-1/4*a^2*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 
 3*c*log(c*x - 1)/d^2) + 1/4*(3*(b^2*c^3*x^3 - b^2*c*x)*arctan2(c*x, sqrt( 
c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 3*(b^2*c^3*x^3 - b^2*c*x)*arctan 
2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(3*b^2*c^2*x^2 - 
2*b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(c^2*d^2*x^3 - d^2 
*x)*integrate(1/2*(4*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (3*( 
b^2*c^4*x^4 - b^2*c^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log( 
c*x + 1) - 3*(b^2*c^4*x^4 - b^2*c^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(- 
c*x + 1))*log(-c*x + 1) - 2*(3*b^2*c^3*x^3 - 2*b^2*c*x)*arctan2(c*x, sqrt( 
c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^4*d^2*x^6 - 2*c 
^2*d^2*x^4 + d^2*x^2), x))/(c^2*d^2*x^3 - d^2*x)
 

3.2.98.8 Giac [F]

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

input

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

output

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

3.2.98.9 Mupad [F(-1)]

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

input

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

output

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