Optimal. Leaf size=41 \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac{\text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]
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Rubi [A] time = 0.080892, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4635, 4406, 3302} \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac{\text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]
Antiderivative was successfully verified.
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Rule 4635
Rule 4406
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^4}{\sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^4(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 x}-\frac{3 \cos (3 x)}{16 x}+\frac{\cos (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{\text{Ci}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Ci}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Ci}\left (5 \sin ^{-1}(a x)\right )}{16 a^5}\\ \end{align*}
Mathematica [A] time = 0.0092898, size = 31, normalized size = 0.76 \[ \frac{2 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )-3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )+\text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{8}}-{\frac{3\,{\it Ci} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{16}}+{\frac{{\it Ci} \left ( 5\,\arcsin \left ( ax \right ) \right ) }{16}} \right ) } \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}}{\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{x^{4}}{\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}}{\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 = 1.3525, size = 47, normalized size = 1.15 \begin{align*} \frac{\operatorname{Ci}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac{3 \, \operatorname{Ci}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} + \frac{\operatorname{Ci}\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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