Optimal. Leaf size=69 \[ -\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}-\frac {5 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a^5}-\frac {x^4 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4631, 3299} \[ -\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}-\frac {5 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a^5}-\frac {x^4 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4631
Rubi steps
\begin {align*} \int \frac {x^4}{\sin ^{-1}(a x)^2} \, dx &=-\frac {x^4 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{8 x}+\frac {9 \sin (3 x)}{16 x}-\frac {5 \sin (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=-\frac {x^4 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^5}-\frac {5 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}+\frac {9 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac {x^4 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}-\frac {5 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a^5}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 61, normalized size = 0.88 \[ -\frac {\frac {16 a^4 x^4 \sqrt {1-a^2 x^2}}{\sin ^{-1}(a x)}+2 \text {Si}\left (\sin ^{-1}(a x)\right )-9 \text {Si}\left (3 \sin ^{-1}(a x)\right )+5 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 115, normalized size = 1.67 \[ -\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{a^{5} \arcsin \left (a x\right )} - \frac {5 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} + \frac {9 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} \arcsin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 81, normalized size = 1.17 \[ \frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )}-\frac {\Si \left (\arcsin \left (a x \right )\right )}{8}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )}+\frac {9 \Si \left (3 \arcsin \left (a x \right )\right )}{16}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )}-\frac {5 \Si \left (5 \arcsin \left (a x \right )\right )}{16}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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