Optimal. Leaf size=38 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4631, 3302} \[ \frac {\text {CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4631
Rubi steps
\begin {align*} \int \frac {x}{\sin ^{-1}(a x)^2} \, dx &=-\frac {x \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac {\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 0.84 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac {\sin \left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 36, normalized size = 0.95 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x}{a \arcsin \left (a x\right )} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.74 \[ \frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{2 \arcsin \left (a x \right )}+\Ci \left (2 \arcsin \left (a x \right )\right )}{a^{2}} \]
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 \[ \frac {-\sqrt {a x + 1} \sqrt {-a x + 1} x + {\left (\int \frac {2 \, \sqrt {-a x + 1} a^{2} x^{2}}{\sqrt {a x + 1} a x \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) - \sqrt {a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} + \int -\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1} a x \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) - \sqrt {a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x}\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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