3.3.63 \(\int \frac {(a+b \arcsin (c x))^2}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\) [263]

3.3.63.1 Optimal result

Integrand size = 29, antiderivative size = 538 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \arcsin (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {32 b c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]

output

-1/3*(a+b*arcsin(c*x))^2/d/x^3/(-c^2*d*x^2+d)^(3/2)-2*c^2*(a+b*arcsin(c*x) 
)^2/d/x/(-c^2*d*x^2+d)^(3/2)+8/3*c^4*x*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d 
)^(3/2)-1/3*b^2*c^2/d^2/x/(-c^2*d*x^2+d)^(1/2)+2/3*b^2*c^4*x/d^2/(-c^2*d*x 
^2+d)^(1/2)+16/3*c^4*x*(a+b*arcsin(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b* 
c*(a+b*arcsin(c*x))/d^2/x^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-16/3*I 
*c^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-32/3* 
b*c^3*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1) 
^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+32/3*b*c^3*(a+b*arcsin(c*x))*ln(1+(I*c*x+( 
-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*I*b^ 
2*c^3*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d^2/(-c^ 
2*d*x^2+d)^(1/2)-8/3*I*b^2*c^3*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c 
^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)
 

3.3.63.2 Mathematica [A] (verified)

Time = 3.37 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {-\frac {a^2 \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )}{x^3}-\frac {a b \left (2 \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \arcsin (c x)+c x \sqrt {1-c^2 x^2} \left (1+16 c^2 x^2 \left (-1+c^2 x^2\right ) \log (c x)+8 c^2 x^2 \left (-1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )\right )\right )}{x^3}+b^2 c^3 \left (1-c^2 x^2\right )^{3/2} \left (\frac {c x}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c x}-\frac {\arcsin (c x)}{c^2 x^2}+\frac {\arcsin (c x)}{-1+c^2 x^2}-16 i \arcsin (c x)^2+\frac {c x \arcsin (c x)^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {8 c x \arcsin (c x)^2}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \arcsin (c x)^2}{c^3 x^3}-\frac {8 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{c x}+16 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+16 \arcsin (c x) \log \left (1+e^{2 i \arcsin (c x)}\right )-8 i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-8 i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}} \]

input

Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]
 

output

(-((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/x^3) - (a*b*(2*(1 + 6*c 
^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x] + c*x*Sqrt[1 - c^2*x^2]*(1 + 
 16*c^2*x^2*(-1 + c^2*x^2)*Log[c*x] + 8*c^2*x^2*(-1 + c^2*x^2)*Log[1 - c^2 
*x^2])))/x^3 + b^2*c^3*(1 - c^2*x^2)^(3/2)*((c*x)/Sqrt[1 - c^2*x^2] - Sqrt 
[1 - c^2*x^2]/(c*x) - ArcSin[c*x]/(c^2*x^2) + ArcSin[c*x]/(-1 + c^2*x^2) - 
 (16*I)*ArcSin[c*x]^2 + (c*x*ArcSin[c*x]^2)/(1 - c^2*x^2)^(3/2) + (8*c*x*A 
rcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - (Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c^3*x 
^3) - (8*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*x) + 16*ArcSin[c*x]*Log[1 - E 
^((2*I)*ArcSin[c*x])] + 16*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (8 
*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (8*I)*PolyLog[2, E^((2*I)*ArcSin[ 
c*x])]))/(3*d*(d - c^2*d*x^2)^(3/2))
 

3.3.63.3 Rubi [A] (verified)

Time = 4.67 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.34, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {5204, 5204, 245, 208, 5162, 5160, 5180, 3042, 4202, 2620, 2715, 2838, 5182, 208, 5208, 208, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 245

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5208

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5184

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

\(\Big \downarrow \) 4919

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]
 

output

-1/3*(a + b*ArcSin[c*x])^2/(d*x^3*(d - c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[1 - 
 c^2*x^2]*((b*c*(-(1/(x*Sqrt[1 - c^2*x^2])) + (2*c^2*x)/Sqrt[1 - c^2*x^2]) 
)/2 - (a + b*ArcSin[c*x])/(2*x^2*(1 - c^2*x^2)) + 2*c^2*(-1/2*(b*c*x)/Sqrt 
[1 - c^2*x^2] + (a + b*ArcSin[c*x])/(2*(1 - c^2*x^2)) + 2*(-((a + b*ArcSin 
[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLog[2, -E^((2*I)*ArcS 
in[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))))/(3*d^2*Sqrt[d - 
c^2*d*x^2]) + 2*c^2*(-((a + b*ArcSin[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2))) 
+ 4*c^2*((x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) - (2*b*c*Sq 
rt[1 - c^2*x^2]*(-1/2*(b*x)/(c*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(2 
*c^2*(1 - c^2*x^2))))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (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*Ar 
cSin[c*x])^2)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*Arc 
Sin[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/4)))/(c*d*Sqrt[d - c^2 
*d*x^2])))/(3*d)) + (2*b*c*Sqrt[1 - c^2*x^2]*(-1/2*(b*c*x)/Sqrt[1 - c^2*x^ 
2] + (a + b*ArcSin[c*x])/(2*(1 - c^2*x^2)) + 2*(-((a + b*ArcSin[c*x])*ArcT 
anh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - 
 (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])))/(d^2*Sqrt[d - c^2*d*x^2]))
 

3.3.63.3.1 Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 245

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

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])
 

3.3.63.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5224 vs. \(2 (517 ) = 1034\).

Time = 0.34 (sec) , antiderivative size = 5225, normalized size of antiderivative = 9.71

method	result	size
default	\(\text {Expression too large to display}\)	\(5225\)
parts	\(\text {Expression too large to display}\)	\(5225\)

input

int((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.3.63.5 Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

input

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),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^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)
 

3.3.63.6 Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

input

integrate((a+b*asin(c*x))**2/x**4/(-c**2*d*x**2+d)**(5/2),x)
 

output

Integral((a + b*asin(c*x))**2/(x**4*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)
 

3.3.63.7 Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

input

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxim 
a")
 

output

1/3*a*b*c*(8*c^2*log(c*x + 1)/d^(5/2) + 8*c^2*log(c*x - 1)/d^(5/2) + 16*c^ 
2*log(x)/d^(5/2) + 1/(c^2*d^(5/2)*x^4 - d^(5/2)*x^2)) + 2/3*(16*c^4*x/(sqr 
t(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2 
*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*a*b*arcsin(c*x) 
 + 1/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3 
/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d* 
x^3))*a^2 + b^2*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c 
^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sq 
rt(d)
 

3.3.63.8 Giac [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

input

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac" 
)
 

output

integrate((b*arcsin(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*x^4), x)
 

3.3.63.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

input

int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^(5/2)),x)
 

output

int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^(5/2)), x)