\(\int \frac {(a+b \arcsin (c x))^2}{x^4 (d-c^2 d x^2)^{3/2}} \, dx\) [250]

Optimal result

Integrand size = 29, antiderivative size = 473 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 d^2 x}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \arcsin (c x))^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arcsin (c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 d \sqrt {d-c^2 d x^2}} \] Output:

-1/3*b^2*c^2*(-c^2*d*x^2+d)^(1/2)/d^2/x-1/3*b*c*(-c^2*x^2+1)^(1/2)*(a+b*ar 
csin(c*x))/d/x^2/(-c^2*d*x^2+d)^(1/2)-1/3*(a+b*arcsin(c*x))^2/d/x^3/(-c^2* 
d*x^2+d)^(1/2)-4/3*c^2*(a+b*arcsin(c*x))^2/d/x/(-c^2*d*x^2+d)^(1/2)+8/3*c^ 
4*x*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)-8/3*I*c^3*(-c^2*x^2+1)^(1/2 
)*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)-20/3*b*c^3*(-c^2*x^2+1)^(1/2) 
*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^ 
(1/2)+16/3*b*c^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^ 
2+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)-I*b^2*c^3*(-c^2*x^2+1)^(1/2)*polylog 
(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)-5/3*I*b^2*c^3*(-c 
^2*x^2+1)^(1/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^( 
1/2)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-a^2-4 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4+b^2 c^4 x^4-a b c x \sqrt {1-c^2 x^2}-2 a b \arcsin (c x)-8 a b c^2 x^2 \arcsin (c x)+16 a b c^4 x^4 \arcsin (c x)-b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x)-b^2 \arcsin (c x)^2-4 b^2 c^2 x^2 \arcsin (c x)^2+8 b^2 c^4 x^4 \arcsin (c x)^2-8 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2+10 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+6 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+e^{2 i \arcsin (c x)}\right )+10 a b c^3 x^3 \sqrt {1-c^2 x^2} \log (c x)+3 a b c^3 x^3 \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )-3 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-5 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 d x^3 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)),x]
 

Output:

(-a^2 - 4*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 + b^2*c^4*x^4 - a*b*c* 
x*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSin[c*x] - 8*a*b*c^2*x^2*ArcSin[c*x] + 16*a 
*b*c^4*x^4*ArcSin[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b^2*ArcSi 
n[c*x]^2 - 4*b^2*c^2*x^2*ArcSin[c*x]^2 + 8*b^2*c^4*x^4*ArcSin[c*x]^2 - (8* 
I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 10*b^2*c^3*x^3*Sqrt[1 - c 
^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 6*b^2*c^3*x^3*Sqrt[1 
- c^2*x^2]*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + 10*a*b*c^3*x^3*Sqr 
t[1 - c^2*x^2]*Log[c*x] + 3*a*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] 
 - (3*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] 
- (5*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/( 
3*d*x^3*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 3.39 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.98, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {5204, 5204, 242, 5160, 5180, 3042, 4202, 2620, 2715, 2838, 5184, 4919, 3042, 4671, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5184

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

\(\Big \downarrow \) 4919

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)),x]
 

Output:

-1/3*(a + b*ArcSin[c*x])^2/(d*x^3*Sqrt[d - c^2*d*x^2]) + (2*b*c*Sqrt[1 - c 
^2*x^2]*(-1/2*(b*c*Sqrt[1 - c^2*x^2])/x - (a + b*ArcSin[c*x])/(2*x^2) + 2* 
c^2*(-((a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyL 
og[2, -E^((2*I)*ArcSin[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]) 
))/(3*d*Sqrt[d - c^2*d*x^2]) + (4*c^2*(-((a + b*ArcSin[c*x])^2/(d*x*Sqrt[d 
 - c^2*d*x^2])) + 2*c^2*((x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) 
 - (2*b*Sqrt[1 - c^2*x^2]*(((I/2)*(a + b*ArcSin[c*x])^2)/b - (2*I)*((-1/2* 
I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])] - (b*PolyLog[2, -E^( 
(2*I)*ArcSin[c*x])])/4)))/(c*d*Sqrt[d - c^2*d*x^2])) + (4*b*c*Sqrt[1 - c^2 
*x^2]*(-((a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*Pol 
yLog[2, -E^((2*I)*ArcSin[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x]) 
]))/(d*Sqrt[d - c^2*d*x^2])))/3
 

Defintions of rubi rules used

rule 242

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] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[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 4919

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
 

rule 5160

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

rule 5180

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

rule 5184

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
x_Symbol] :> Simp[1/d   Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi 
n[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2843 vs. \(2 (460 ) = 920\).

Time = 0.89 (sec) , antiderivative size = 2844, normalized size of antiderivative = 6.01

method	result	size
default	\(\text {Expression too large to display}\)	\(2844\)
parts	\(\text {Expression too large to display}\)	\(2844\)

Input:

int((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-128/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^2*(-c^2* 
x^2+1)^(1/2)*arcsin(c*x)*c^5+32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7* 
c^2*x^2-1)/d^2*x^7*c^10-40/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x 
^2-1)/d^2*x^5*c^8+7/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d 
^2*x*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x*c^2+ 
1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^3*arcsin(c*x) 
^2+a^2*(-1/3/d/x^3/(-c^2*d*x^2+d)^(1/2)+4/3*c^2*(-1/d/x/(-c^2*d*x^2+d)^(1/ 
2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2)))+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4* 
x^4-7*c^2*x^2-1)/d^2*(-c^2*x^2+1)^(1/2)*c^3+2/3*a*b*(-d*(c^2*x^2-1))^(1/2) 
/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^3*arcsin(c*x)+64/3*I*a*b*(-d*(c^2*x^2-1))^( 
1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^5*(-c^2*x^2+1)*c^8-32/3*I*a*b*(-d*(c^2* 
x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*(-c^2*x^2+1)*c^6+32/3*I*a*b* 
(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)*c^3- 
8/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*(-c^2*x^2+1 
)*c^4-16/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*(-c^2* 
x^2+1)^(1/2)*arcsin(c*x)*c^3+64/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4- 
7*c^2*x^2-1)/d^2*x^5*(-c^2*x^2+1)*arcsin(c*x)*c^8-32/3*I*b^2*(-d*(c^2*x^2- 
1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*(-c^2*x^2+1)*arcsin(c*x)*c^6-64/ 
3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^2*(-c^2*x^2+1 
)^(1/2)*arcsin(c*x)^2*c^5-8/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7...
 

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b*asin(c*x))**2/x**4/(-c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral((a + b*asin(c*x))**2/(x**4*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)
 

Maxima [F]

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

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxim 
a")
 

Output:

1/3*(8*c^4*x/(sqrt(-c^2*d*x^2 + d)*d) - 4*c^2/(sqrt(-c^2*d*x^2 + d)*d*x) - 
 1/(sqrt(-c^2*d*x^2 + d)*d*x^3))*a^2 + sqrt(d)*integrate((b^2*arctan2(c*x, 
 sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(- 
c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2 
*x^4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(3/2),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. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^(3/2)),x)
 

Output:

int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}-\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) a b \,x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}-\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b^{2} x^{3}+8 a^{2} c^{4} x^{4}-4 a^{2} c^{2} x^{2}-a^{2}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d \,x^{3}} \] Input:

int((a+b*asin(c*x))^2/x^4/(-c^2*d*x^2+d)^(3/2),x)
 

Output:

( - 6*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x* 
*6 - sqrt( - c**2*x**2 + 1)*x**4),x)*a*b*x**3 - 3*sqrt( - c**2*x**2 + 1)*i 
nt(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**2*x**6 - sqrt( - c**2*x**2 + 1) 
*x**4),x)*b**2*x**3 + 8*a**2*c**4*x**4 - 4*a**2*c**2*x**2 - a**2)/(3*sqrt( 
d)*sqrt( - c**2*x**2 + 1)*d*x**3)