Optimal. Leaf size=55 \[ \frac{5 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a^7} \]
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Rubi [A] time = 0.0965148, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4635, 4406, 3302} \[ \frac{5 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a^7} \]
Antiderivative was successfully verified.
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Rule 4635
Rule 4406
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^6}{\sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^6(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \cos (x)}{64 x}-\frac{9 \cos (3 x)}{64 x}+\frac{5 \cos (5 x)}{64 x}-\frac{\cos (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\cos (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}\\ &=\frac{5 \text{Ci}\left (\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{Ci}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \text{Ci}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{\text{Ci}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}\\ \end{align*}
Mathematica [A] time = 0.0164933, size = 40, normalized size = 0.73 \[ -\frac{-5 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )+9 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )-5 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )+\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 40, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{7}} \left ({\frac{5\,{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{64}}-{\frac{9\,{\it Ci} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{64}}+{\frac{5\,{\it Ci} \left ( 5\,\arcsin \left ( ax \right ) \right ) }{64}}-{\frac{{\it Ci} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\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^{6}}{\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^{6}}{\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 = 1.31765, size = 63, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Ci}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac{5 \, \operatorname{Ci}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac{9 \, \operatorname{Ci}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac{5 \, \operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{64 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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