3.3.75 \(\int \frac {\arcsin (a x)^2}{(c-a^2 c x^2)^{7/2}} \, dx\) [275]

3.3.75.1 Optimal result

Integrand size = 22, antiderivative size = 390 \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\arcsin (a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \arcsin (a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \arcsin (a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \arcsin (a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \arcsin (a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]

output

1/5*x*arcsin(a*x)^2/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arcsin(a*x)^2/c^2/(-a^2* 
c*x^2+c)^(3/2)+1/3*x/c^3/(-a^2*c*x^2+c)^(1/2)+1/30*x/c^3/(-a^2*x^2+1)/(-a^ 
2*c*x^2+c)^(1/2)-1/10*arcsin(a*x)/a/c^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^ 
(1/2)+8/15*x*arcsin(a*x)^2/c^3/(-a^2*c*x^2+c)^(1/2)-4/15*arcsin(a*x)/a/c^3 
/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)-8/15*I*arcsin(a*x)^2*(-a^2*x^2+1) 
^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)+16/15*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+ 
1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)-8/15*I*polylog( 
2,-(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^( 
1/2)
 

3.3.75.2 Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.60 \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (\frac {a^3 x^3}{\left (1-a^2 x^2\right )^{3/2}}+\frac {11 a x}{\sqrt {1-a^2 x^2}}-16 i \arcsin (a x)^2+\frac {16 a x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}+\frac {8 \arcsin (a x) \left (-1+\frac {a x \arcsin (a x)}{\sqrt {1-a^2 x^2}}\right )}{1-a^2 x^2}+\frac {3 \arcsin (a x) \left (-1+\frac {2 a x \arcsin (a x)}{\sqrt {1-a^2 x^2}}\right )}{\left (1-a^2 x^2\right )^2}+32 \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )-16 i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )\right )}{30 a c^3 \sqrt {c \left (1-a^2 x^2\right )}} \]

input

Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(7/2),x]
 

output

(Sqrt[1 - a^2*x^2]*((a^3*x^3)/(1 - a^2*x^2)^(3/2) + (11*a*x)/Sqrt[1 - a^2* 
x^2] - (16*I)*ArcSin[a*x]^2 + (16*a*x*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + ( 
8*ArcSin[a*x]*(-1 + (a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2]))/(1 - a^2*x^2) + 
(3*ArcSin[a*x]*(-1 + (2*a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2]))/(1 - a^2*x^2) 
^2 + 32*ArcSin[a*x]*Log[1 + E^((2*I)*ArcSin[a*x])] - (16*I)*PolyLog[2, -E^ 
((2*I)*ArcSin[a*x])]))/(30*a*c^3*Sqrt[c*(1 - a^2*x^2)])
 

3.3.75.3 Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5162, 5162, 5160, 5180, 3042, 4202, 2620, 2715, 2838, 5182, 208, 209, 208}

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.

\(\displaystyle \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

input

Int[ArcSin[a*x]^2/(c - a^2*c*x^2)^(7/2),x]
 

output

(x*ArcSin[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) - (2*a*Sqrt[1 - a^2*x^2]*(-1 
/4*(x/(3*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2]))/a + ArcSin[a* 
x]/(4*a^2*(1 - a^2*x^2)^2)))/(5*c^3*Sqrt[c - a^2*c*x^2]) + (4*((x*ArcSin[a 
*x]^2)/(3*c*(c - a^2*c*x^2)^(3/2)) - (2*a*Sqrt[1 - a^2*x^2]*(-1/2*x/(a*Sqr 
t[1 - a^2*x^2]) + ArcSin[a*x]/(2*a^2*(1 - a^2*x^2))))/(3*c^2*Sqrt[c - a^2* 
c*x^2]) + (2*((x*ArcSin[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[1 - a^2* 
x^2]*((I/2)*ArcSin[a*x]^2 - (2*I)*((-1/2*I)*ArcSin[a*x]*Log[1 + E^((2*I)*A 
rcSin[a*x])] - PolyLog[2, -E^((2*I)*ArcSin[a*x])]/4)))/(a*c*Sqrt[c - a^2*c 
*x^2])))/(3*c)))/(5*c)
 

3.3.75.3.1 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 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

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 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]
 

3.3.75.4 Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.43

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+15 a x -16 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 i \sqrt {-a^{2} x^{2}+1}\right ) \left (-156 i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+64 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+456 i \arcsin \left (a x \right ) a^{4} x^{4}+32 a^{8} x^{8}+126 i \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-248 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+64 i \arcsin \left (a x \right ) a^{8} x^{8}-142 a^{6} x^{6}+80 a^{4} x^{4} \arcsin \left (a x \right )^{2}+88 i \arcsin \left (a x \right )+340 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-280 i \arcsin \left (a x \right ) a^{6} x^{6}+265 a^{4} x^{4}-190 \arcsin \left (a x \right )^{2} a^{2} x^{2}-328 i \arcsin \left (a x \right ) a^{2} x^{2}-165 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +62 i \sqrt {-a^{2} x^{2}+1}\, a x -235 a^{2} x^{2}+128 \arcsin \left (a x \right )^{2}-32 i \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+80\right )}{30 c^{4} \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) a}+\frac {8 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\operatorname {polylog}\left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{15 c^{4} \left (a^{2} x^{2}-1\right ) a}\)	\(556\)

input

int(arcsin(a*x)^2/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
 

output

-1/30*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-20*a^3*x^3+8*I*(-a^2*x^2+1)^(1/2)* 
a^4*x^4+15*a*x-16*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+8*I*(-a^2*x^2+1)^(1/2))*(-1 
56*I*(-a^2*x^2+1)^(1/2)*a^3*x^3+64*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^7*x^7+ 
456*I*arcsin(a*x)*a^4*x^4+32*a^8*x^8+126*I*(-a^2*x^2+1)^(1/2)*a^5*x^5-248* 
arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^5*x^5+64*I*arcsin(a*x)*a^8*x^8-142*a^6*x^ 
6+80*a^4*x^4*arcsin(a*x)^2+88*I*arcsin(a*x)+340*arcsin(a*x)*(-a^2*x^2+1)^( 
1/2)*a^3*x^3-280*I*arcsin(a*x)*a^6*x^6+265*a^4*x^4-190*arcsin(a*x)^2*a^2*x 
^2-328*I*arcsin(a*x)*a^2*x^2-165*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x+62*I*( 
-a^2*x^2+1)^(1/2)*a*x-235*a^2*x^2+128*arcsin(a*x)^2-32*I*(-a^2*x^2+1)^(1/2 
)*a^7*x^7+80)/c^4/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x^4+287*a^ 
2*x^2-64)/a+8/15*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(2*I*arcsin(a 
*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+2*arcsin(a*x)^2+polylog(2,-(I*a*x+( 
-a^2*x^2+1)^(1/2))^2))/c^4/(a^2*x^2-1)/a
 

3.3.75.5 Fricas [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

input

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")
 

output

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

3.3.75.6 Sympy [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]

input

integrate(asin(a*x)**2/(-a**2*c*x**2+c)**(7/2),x)
 

output

Integral(asin(a*x)**2/(-c*(a*x - 1)*(a*x + 1))**(7/2), x)
 

3.3.75.7 Maxima [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

input

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")
 

output

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(7/2), x)
 

3.3.75.8 Giac [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

input

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")
 

output

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(7/2), x)
 

3.3.75.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]

input

int(asin(a*x)^2/(c - a^2*c*x^2)^(7/2),x)
 

output

int(asin(a*x)^2/(c - a^2*c*x^2)^(7/2), x)