Optimal. Leaf size=41 \[ -\frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^5}+\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{8 a^5}+\frac{3 \log \left (\sin ^{-1}(a x)\right )}{8 a^5} \]
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Rubi [A] time = 0.158949, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4723, 3312, 3302} \[ -\frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^5}+\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{8 a^5}+\frac{3 \log \left (\sin ^{-1}(a x)\right )}{8 a^5} \]
Antiderivative was successfully verified.
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Rule 4723
Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin ^4(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 x}-\frac{\cos (2 x)}{2 x}+\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=\frac{3 \log \left (\sin ^{-1}(a x)\right )}{8 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{\text{Ci}\left (2 \sin ^{-1}(a x)\right )}{2 a^5}+\frac{\text{Ci}\left (4 \sin ^{-1}(a x)\right )}{8 a^5}+\frac{3 \log \left (\sin ^{-1}(a x)\right )}{8 a^5}\\ \end{align*}
Mathematica [A] time = 0.0718342, size = 31, normalized size = 0.76 \[ \frac{-4 \text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )+\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )+3 \log \left (\sin ^{-1}(a x)\right )}{8 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2\,{a}^{5}}}+{\frac{{\it Ci} \left ( 4\,\arcsin \left ( ax \right ) \right ) }{8\,{a}^{5}}}+{\frac{3\,\ln \left ( \arcsin \left ( ax \right ) \right ) }{8\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}\,{d x} \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{\sqrt{-a^{2} x^{2} + 1} x^{4}}{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}, 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^{4}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{asin}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31518, size = 47, normalized size = 1.15 \begin{align*} \frac{\operatorname{Ci}\left (4 \, \arcsin \left (a x\right )\right )}{8 \, a^{5}} - \frac{\operatorname{Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{5}} + \frac{3 \, \log \left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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