\(\int \frac {(d-c^2 d x^2)^3 (a+b \arcsin (c x))}{x} \, dx\) [24]

Optimal result

Integrand size = 25, antiderivative size = 235 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \arcsin (c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))-\frac {i d^3 (a+b \arcsin (c x))^2}{2 b}+d^3 (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i b d^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \] Output:

-19/48*b*c*d^3*x*(-c^2*x^2+1)^(1/2)-7/72*b*c*d^3*x*(-c^2*x^2+1)^(3/2)-1/36 
*b*c*d^3*x*(-c^2*x^2+1)^(5/2)-19/48*b*d^3*arcsin(c*x)+1/2*d^3*(-c^2*x^2+1) 
*(a+b*arcsin(c*x))+1/4*d^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))+1/6*d^3*(-c^2* 
x^2+1)^3*(a+b*arcsin(c*x))-1/2*I*d^3*(a+b*arcsin(c*x))^2/b+d^3*(a+b*arcsin 
(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*I*b*d^3*polylog(2,(I*c*x+(-c 
^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=-\frac {1}{144} d^3 \left (216 a c^2 x^2-108 a c^4 x^4+24 a c^6 x^6+75 b c x \sqrt {1-c^2 x^2}-22 b c^3 x^3 \sqrt {1-c^2 x^2}+4 b c^5 x^5 \sqrt {1-c^2 x^2}+72 i b \arcsin (c x)^2-150 b \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )+12 b \arcsin (c x) \left (18 c^2 x^2-9 c^4 x^4+2 c^6 x^6-12 \log \left (1-e^{2 i \arcsin (c x)}\right )\right )-144 a \log (x)+72 i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \] Input:

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

Output:

-1/144*(d^3*(216*a*c^2*x^2 - 108*a*c^4*x^4 + 24*a*c^6*x^6 + 75*b*c*x*Sqrt[ 
1 - c^2*x^2] - 22*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 4*b*c^5*x^5*Sqrt[1 - c^2*x 
^2] + (72*I)*b*ArcSin[c*x]^2 - 150*b*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2]) 
] + 12*b*ArcSin[c*x]*(18*c^2*x^2 - 9*c^4*x^4 + 2*c^6*x^6 - 12*Log[1 - E^(( 
2*I)*ArcSin[c*x])]) - 144*a*Log[x] + (72*I)*b*PolyLog[2, E^((2*I)*ArcSin[c 
*x])]))
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.43, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {5188, 27, 211, 211, 211, 223, 5188, 211, 211, 223, 5188, 211, 223, 5136, 3042, 25, 4200, 25, 2620, 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.

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

\(\Big \downarrow \) 5188

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5188

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5188

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle d^3 \left (\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )\)

Input:

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

Output:

(d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/6 - (b*c*d^3*((x*(1 - c^2*x^2)^( 
5/2))/6 + (5*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + Ar 
cSin[c*x]/(2*c)))/4))/6))/6 + d^3*(((1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 + 
 ((1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/4 - ((I/2)*(a + b*ArcSin[c*x])^2)/b 
 - (b*c*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/2 - (b*c*((x*(1 - c 
^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/4 
 - (2*I)*((I/2)*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] + (b*Po 
lyLog[2, E^((2*I)*ArcSin[c*x])])/4))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

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

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( 
a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
 

rule 5188

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.94

method	result	size
parts	\(-d^{3} a \left (\frac {c^{6} x^{6}}{6}-\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\ln \left (x \right )\right )-d^{3} b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}+\frac {\sin \left (6 \arcsin \left (c x \right )\right )}{1152}-\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {29 \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}+\frac {29 \sin \left (2 \arcsin \left (c x \right )\right )}{128}\right )\)	\(221\)
derivativedivides	\(-d^{3} a \left (\frac {c^{6} x^{6}}{6}-\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-d^{3} b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}+\frac {\sin \left (6 \arcsin \left (c x \right )\right )}{1152}-\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {29 \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}+\frac {29 \sin \left (2 \arcsin \left (c x \right )\right )}{128}\right )\)	\(223\)
default	\(-d^{3} a \left (\frac {c^{6} x^{6}}{6}-\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-d^{3} b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}+\frac {\sin \left (6 \arcsin \left (c x \right )\right )}{1152}-\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {29 \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}+\frac {29 \sin \left (2 \arcsin \left (c x \right )\right )}{128}\right )\)	\(223\)

Input:

int((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))/x,x,method=_RETURNVERBOSE)
 

Output:

-d^3*a*(1/6*c^6*x^6-3/4*c^4*x^4+3/2*c^2*x^2-ln(x))-d^3*b*(1/2*I*arcsin(c*x 
)^2-arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+I*polylog(2,I*c*x+(-c^2*x^2 
+1)^(1/2))-arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+I*polylog(2,-I*c*x-( 
-c^2*x^2+1)^(1/2))-1/192*arcsin(c*x)*cos(6*arcsin(c*x))+1/1152*sin(6*arcsi 
n(c*x))-1/16*arcsin(c*x)*cos(4*arcsin(c*x))+1/64*sin(4*arcsin(c*x))-29/64* 
arcsin(c*x)*cos(2*arcsin(c*x))+29/128*sin(2*arcsin(c*x)))
 

Fricas [F]

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

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

Output:

integral(-(a*c^6*d^3*x^6 - 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 - a*d^3 + (b* 
c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*c^2*d^3*x^2 - b*d^3)*arcsin(c*x))/x, x 
)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=- d^{3} \left (\int \left (- \frac {a}{x}\right )\, dx + \int 3 a c^{2} x\, dx + \int \left (- 3 a c^{4} x^{3}\right )\, dx + \int a c^{6} x^{5}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int 3 b c^{2} x \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{3} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{5} \operatorname {asin}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(-a/x, x) + Integral(3*a*c**2*x, x) + Integral(-3*a*c**4*x* 
*3, x) + Integral(a*c**6*x**5, x) + Integral(-b*asin(c*x)/x, x) + Integral 
(3*b*c**2*x*asin(c*x), x) + Integral(-3*b*c**4*x**3*asin(c*x), x) + Integr 
al(b*c**6*x**5*asin(c*x), x))
 

Maxima [F]

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

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

Output:

-1/6*a*c^6*d^3*x^6 + 3/4*a*c^4*d^3*x^4 - 3/2*a*c^2*d^3*x^2 + a*d^3*log(x) 
- integrate((b*c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*c^2*d^3*x^2 - b*d^3)*ar 
ctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))}{x} \, dx=\frac {d^{3} \left (-24 \mathit {asin} \left (c x \right ) b \,c^{6} x^{6}+108 \mathit {asin} \left (c x \right ) b \,c^{4} x^{4}-216 \mathit {asin} \left (c x \right ) b \,c^{2} x^{2}+75 \mathit {asin} \left (c x \right ) b -4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} x^{5}+22 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} x^{3}-75 \sqrt {-c^{2} x^{2}+1}\, b c x +144 \left (\int \frac {\mathit {asin} \left (c x \right )}{x}d x \right ) b +144 \,\mathrm {log}\left (x \right ) a -24 a \,c^{6} x^{6}+108 a \,c^{4} x^{4}-216 a \,c^{2} x^{2}\right )}{144} \] Input:

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

Output:

(d**3*( - 24*asin(c*x)*b*c**6*x**6 + 108*asin(c*x)*b*c**4*x**4 - 216*asin( 
c*x)*b*c**2*x**2 + 75*asin(c*x)*b - 4*sqrt( - c**2*x**2 + 1)*b*c**5*x**5 + 
 22*sqrt( - c**2*x**2 + 1)*b*c**3*x**3 - 75*sqrt( - c**2*x**2 + 1)*b*c*x + 
 144*int(asin(c*x)/x,x)*b + 144*log(x)*a - 24*a*c**6*x**6 + 108*a*c**4*x** 
4 - 216*a*c**2*x**2))/144