Optimal. Leaf size=388 \[ -\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \text {Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4655, 4653, 4675, 3719, 2190, 2531, 2282, 6589, 4677, 4651, 260} \[ -\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \text {PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2190
Rule 2282
Rule 2531
Rule 3719
Rule 4651
Rule 4653
Rule 4655
Rule 4675
Rule 4677
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (2 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (4 i \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (4 \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (2 i \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \text {Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 211, normalized size = 0.54 \[ \frac {\left (1-a^2 x^2\right )^{3/2} \left (3 \log \left (1-a^2 x^2\right )+\frac {4 a x \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}+\frac {2 a x \sin ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}+\frac {3 \sin ^{-1}(a x)^2}{a^2 x^2-1}+\frac {6 a x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-12 i \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )+6 \text {Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )-4 i \sin ^{-1}(a x)^3+12 \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )\right )}{6 a c \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 661, normalized size = 1.70 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, x^{2} a^{2}+2 a^{3} x^{3}-2 i \sqrt {-a^{2} x^{2}+1}-3 a x \right ) \arcsin \left (a x \right ) \left (-6 i \arcsin \left (a x \right ) x^{4} a^{4}-6 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, x^{3} a^{3}+6 i \sqrt {-a^{2} x^{2}+1}\, x^{3} a^{3}-6 a^{4} x^{4}+6 \arcsin \left (a x \right )^{2} x^{2} a^{2}+12 i \arcsin \left (a x \right ) x^{2} a^{2}+9 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, x a -6 i \sqrt {-a^{2} x^{2}+1}\, x a +18 a^{2} x^{2}-8 \arcsin \left (a x \right )^{2}-6 i \arcsin \left (a x \right )-12\right )}{6 c^{3} \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) a}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}+\frac {4 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \arcsin \left (a x \right )^{3}}{3 a \,c^{3} \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{2} \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \arcsin \left (a x \right ) \polylog \left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \polylog \left (3, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 106, normalized size = 0.27 \[ \frac {1}{2} \, a {\left (\frac {1}{a^{4} c^{\frac {5}{2}} x^{2} - a^{2} c^{\frac {5}{2}}} + \frac {2 \, \log \left (a x + 1\right )}{a^{2} c^{\frac {5}{2}}} + \frac {2 \, \log \left (a x - 1\right )}{a^{2} c^{\frac {5}{2}}}\right )} \arcsin \left (a x\right )^{2} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \arcsin \left (a x\right )^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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