Optimal. Leaf size=547 \[ -\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \text {Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.48, antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {4655, 4653, 4675, 3719, 2190, 2531, 2282, 6589, 4677, 4651, 260, 261} \[ -\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \text {PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
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 )^{7/2}} \, dx &=\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (3 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}+\frac {\left (3 \sqrt {1-a^2 x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (4 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}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \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}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {1-a^2 x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{10 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (8 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (8 \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{5 a c^3 \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}{5 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (4 a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \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^3 \sqrt {c-a^2 c x^2}}+\frac {\left (16 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 )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {\left (16 \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 )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 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 )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \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^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \text {Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 319, normalized size = 0.58 \[ \frac {-96 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )+48 \sqrt {1-a^2 x^2} \text {Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )-\frac {3}{\sqrt {1-a^2 x^2}}+30 \sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )-32 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3+\frac {16 a x \sin ^{-1}(a x)^3}{1-a^2 x^2}+\frac {12 a x \sin ^{-1}(a x)^3}{\left (a^2 x^2-1\right )^2}-\frac {24 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {9 \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}+\frac {6 a x \sin ^{-1}(a x)}{1-a^2 x^2}+96 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )+32 a x \sin ^{-1}(a x)^3+60 a x \sin ^{-1}(a x)}{60 a c^3 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 1017, normalized size = 1.86 \[ -\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}\, x^{4} a^{4}+15 a x -16 i \sqrt {-a^{2} x^{2}+1}\, x^{2} a^{2}+8 i \sqrt {-a^{2} x^{2}+1}\right ) \left (24 i x^{8} a^{8}-96 i x^{6} a^{6}+144 i x^{4} a^{4}+192 \arcsin \left (a x \right ) x^{8} a^{8}-852 \arcsin \left (a x \right ) x^{6} a^{6}-380 \arcsin \left (a x \right )^{3} x^{2} a^{2}+756 i \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right ) x^{5} a^{5}-936 i \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right ) x^{3} a^{3}+372 i \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right ) x a -192 i \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right ) x^{7} a^{7}+1368 i \arcsin \left (a x \right )^{2} x^{4} a^{4}-984 i \arcsin \left (a x \right )^{2} x^{2} a^{2}+192 i \arcsin \left (a x \right )^{2} x^{8} a^{8}-840 i \arcsin \left (a x \right )^{2} x^{6} a^{6}+192 \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{2} x^{7} a^{7}-744 \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{2} x^{5} a^{5}+264 i \arcsin \left (a x \right )^{2}+256 \arcsin \left (a x \right )^{3}+480 \arcsin \left (a x \right )+1590 a^{4} x^{4} \arcsin \left (a x \right )+160 a^{4} x^{4} \arcsin \left (a x \right )^{3}-45 a x \sqrt {-a^{2} x^{2}+1}+105 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-1410 a^{2} x^{2} \arcsin \left (a x \right )+24 i+24 \sqrt {-a^{2} x^{2}+1}\, x^{7} a^{7}-96 i x^{2} a^{2}+1020 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, x^{3} a^{3}-495 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, x a -84 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}\right )}{60 c^{4} \left (40 a^{10} x^{10}-215 x^{8} a^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\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^{4} \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^{4} \left (a^{2} x^{2}-1\right )}+\frac {16 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \arcsin \left (a x \right )^{3}}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {8 \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 )}{5 a \,c^{4} \left (a^{2} x^{2}-1\right )}+\frac {8 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 )}{5 a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {4 \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 )}{5 a \,c^{4} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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