Optimal. Leaf size=105 \[ \frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {8 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4621, 4719, 4723, 3305, 3351} \[ \frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {8 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4621
Rule 4719
Rule 4723
Rubi steps
\begin {align*} \int \frac {1}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {1}{5} (2 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}}-\frac {4}{15} \int \frac {1}{\sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {1}{15} (8 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{15 a}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {8 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 143, normalized size = 1.36 \[ \frac {2 e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )-8 \sqrt {-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (8 \sin ^{-1}(a x)^2+4 i \sin ^{-1}(a x)+8 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )-6\right )}{30 a \sin ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\arcsin \left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 110, normalized size = 1.05 \[ \frac {\sqrt {2}\, \left (8 \arcsin \left (a x \right )^{3} \pi \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+4 \arcsin \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+2 \arcsin \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x a -3 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{15 a \sqrt {\pi }\, \arcsin \left (a x \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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