Optimal. Leaf size=41 \[ \frac {\text {Ci}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac {3 \text {Ci}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac {\text {Ci}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]
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Rubi [A] time = 0.08, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 3302
Rule 4406
Rule 4635
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*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.76 \[ \frac {2 \text {Ci}\left (\sin ^{-1}(a x)\right )-3 \text {Ci}\left (3 \sin ^{-1}(a x)\right )+\text {Ci}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\arcsin \left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 35, normalized size = 0.85 \[ \frac {\operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac {3 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} + \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.76 \[ \frac {\frac {\Ci \left (\arcsin \left (a x \right )\right )}{8}-\frac {3 \Ci \left (3 \arcsin \left (a x \right )\right )}{16}+\frac {\Ci \left (5 \arcsin \left (a x \right )\right )}{16}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\arcsin \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^4}{\mathrm {asin}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\operatorname {asin}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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