Optimal. Leaf size=191 \[ \frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}-\frac{6 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.347694, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4633, 4719, 4631, 3305, 3351, 4621, 4723} \[ \frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}-\frac{6 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4633
Rule 4719
Rule 4631
Rule 3305
Rule 3351
Rule 4621
Rule 4723
Rubi steps
\begin{align*} \int \frac{x^2}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-\frac{1}{5} (6 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{12}{5} \int \frac{x^2}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac{8 \int \frac{1}{\sin ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{24 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 \sqrt{x}}+\frac{3 \sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}-\frac{16 \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{15 a}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}-\frac{18 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{32 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}+\frac{12 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}-\frac{36 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}-\frac{6 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}\\ \end{align*}
Mathematica [C] time = 0.47627, size = 280, normalized size = 1.47 \[ \frac{-4 \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )+36 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (4 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )+4 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)-3\right )-3 e^{-3 i \sin ^{-1}(a x)} \left (12 \sqrt{3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )+12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)-1\right )+3 e^{3 i \sin ^{-1}(a x)} \left (-12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+1\right )+e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )}{60 a^3 \sin ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.055, size = 154, normalized size = 0.8 \begin{align*} -{\frac{1}{30\,{a}^{3}} \left ( 36\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}-4\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}-4\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+36\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) -2\,ax\arcsin \left ( ax \right ) +6\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) +3\,\sqrt{-{a}^{2}{x}^{2}+1}-3\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\arcsin \left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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