Optimal. Leaf size=171 \[ -\frac{5 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{9 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{\sqrt{\frac{7 \pi }{2}} S\left (\sqrt{\frac{14}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}-\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.14418, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4631, 3305, 3351} \[ -\frac{5 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{9 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{\sqrt{\frac{7 \pi }{2}} S\left (\sqrt{\frac{14}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}-\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4631
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^6}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{5 \sin (x)}{64 \sqrt{x}}+\frac{27 \sin (3 x)}{64 \sqrt{x}}-\frac{25 \sin (5 x)}{64 \sqrt{x}}+\frac{7 \sin (7 x)}{64 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=-\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^7}+\frac{7 \operatorname{Subst}\left (\int \frac{\sin (7 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^7}-\frac{25 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^7}+\frac{27 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^7}\\ &=-\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{5 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{7 \operatorname{Subst}\left (\int \sin \left (7 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}-\frac{25 \operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{27 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}\\ &=-\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{5 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{9 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}+\frac{\sqrt{\frac{7 \pi }{2}} S\left (\sqrt{\frac{14}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^7}\\ \end{align*}
Mathematica [C] time = 0.233579, size = 427, normalized size = 2.5 \[ \frac{-\frac{5 \left (e^{i \sin ^{-1}(a x)}-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )\right )}{64 \sqrt{\sin ^{-1}(a x)}}-\frac{5 \left (e^{-i \sin ^{-1}(a x)}-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )\right )}{64 \sqrt{\sin ^{-1}(a x)}}+\frac{9 \left (e^{3 i \sin ^{-1}(a x)}-\sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )\right )}{64 \sqrt{\sin ^{-1}(a x)}}+\frac{9 \left (e^{-3 i \sin ^{-1}(a x)}-\sqrt{3} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )\right )}{64 \sqrt{\sin ^{-1}(a x)}}-\frac{5 \left (e^{5 i \sin ^{-1}(a x)}-\sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 i \sin ^{-1}(a x)\right )\right )}{64 \sqrt{\sin ^{-1}(a x)}}-\frac{5 \left (e^{-5 i \sin ^{-1}(a x)}-\sqrt{5} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 i \sin ^{-1}(a x)\right )\right )}{64 \sqrt{\sin ^{-1}(a x)}}+\frac{e^{7 i \sin ^{-1}(a x)}-\sqrt{7} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-7 i \sin ^{-1}(a x)\right )}{64 \sqrt{\sin ^{-1}(a x)}}+\frac{e^{-7 i \sin ^{-1}(a x)}-\sqrt{7} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},7 i \sin ^{-1}(a x)\right )}{64 \sqrt{\sin ^{-1}(a x)}}}{a^7} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.077, size = 184, normalized size = 1.1 \begin{align*} -{\frac{1}{32\,{a}^{7}} \left ( -\sqrt{2}\sqrt{\pi }\sqrt{7}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{7}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) \sqrt{\arcsin \left ( ax \right ) }+5\,\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -9\,\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +5\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +5\,\sqrt{-{a}^{2}{x}^{2}+1}-9\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) +5\,\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) -\cos \left ( 7\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}}\, dx \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^{6}}{\arcsin \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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