Optimal. Leaf size=83 \[ -\frac{5 \text{Si}\left (\sin ^{-1}(a x)\right )}{64 a^7}+\frac{27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{25 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{7 \text{Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{x^6 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.0744056, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4631, 3299} \[ -\frac{5 \text{Si}\left (\sin ^{-1}(a x)\right )}{64 a^7}+\frac{27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{25 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{7 \text{Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{x^6 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4631
Rule 3299
Rubi steps
\begin{align*} \int \frac{x^6}{\sin ^{-1}(a x)^2} \, dx &=-\frac{x^6 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{5 \sin (x)}{64 x}+\frac{27 \sin (3 x)}{64 x}-\frac{25 \sin (5 x)}{64 x}+\frac{7 \sin (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=-\frac{x^6 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac{7 \operatorname{Subst}\left (\int \frac{\sin (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}-\frac{25 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac{27 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}\\ &=-\frac{x^6 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{5 \text{Si}\left (\sin ^{-1}(a x)\right )}{64 a^7}+\frac{27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{25 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{7 \text{Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}\\ \end{align*}
Mathematica [A] time = 0.244355, size = 86, normalized size = 1.04 \[ -\frac{64 a^6 x^6 \sqrt{1-a^2 x^2}+5 \sin ^{-1}(a x) \text{Si}\left (\sin ^{-1}(a x)\right )-27 \sin ^{-1}(a x) \text{Si}\left (3 \sin ^{-1}(a x)\right )+25 \sin ^{-1}(a x) \text{Si}\left (5 \sin ^{-1}(a x)\right )-7 \sin ^{-1}(a x) \text{Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7 \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 105, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{7}} \left ( -{\frac{5}{64\,\arcsin \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{5\,{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{64}}+{\frac{9\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) }{64\,\arcsin \left ( ax \right ) }}+{\frac{27\,{\it Si} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{64}}-{\frac{5\,\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) }{64\,\arcsin \left ( ax \right ) }}-{\frac{25\,{\it Si} \left ( 5\,\arcsin \left ( ax \right ) \right ) }{64}}+{\frac{\cos \left ( 7\,\arcsin \left ( ax \right ) \right ) }{64\,\arcsin \left ( ax \right ) }}+{\frac{7\,{\it Si} \left ( 7\,\arcsin \left ( ax \right ) \right ) }{64}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{6}}{\arcsin \left (a x\right )^{2}}, x\right ) \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}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28651, size = 217, normalized size = 2.61 \begin{align*} -\frac{{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt{-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} - \frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} + \frac{7 \, \operatorname{Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac{25 \, \operatorname{Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac{27 \, \operatorname{Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac{5 \, \operatorname{Si}\left (\arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac{3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a^{7} \arcsin \left (a x\right )} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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