Optimal. Leaf size=55 \[ -\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4631, 3299} \[ -\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4631
Rubi steps
\begin {align*} \int \frac {x^2}{\sin ^{-1}(a x)^2} \, dx &=-\frac {x^2 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{4 x}+\frac {3 \sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2 \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 )}{4 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 50, normalized size = 0.91 \[ -\frac {\frac {4 a^2 x^2 \sqrt {1-a^2 x^2}}{\sin ^{-1}(a x)}+\text {Si}\left (\sin ^{-1}(a x)\right )-3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 68, normalized size = 1.24 \[ \frac {3 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a^{3}} - \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{4 \, a^{3}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{3} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} \arcsin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 1.04 \[ \frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{4 \arcsin \left (a x \right )}-\frac {\Si \left (\arcsin \left (a x \right )\right )}{4}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )}+\frac {3 \Si \left (3 \arcsin \left (a x \right )\right )}{4}}{a^{3}} \]
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 = -0.02 \[ \int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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