Optimal. Leaf size=32 \[ \frac {\sin ^{-1}(a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5994, 216} \[ \frac {\sin ^{-1}(a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^2}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {\sin ^{-1}(a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 0.91 \[ \frac {\sin ^{-1}(a x)-\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 58, normalized size = 1.81 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 4 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 47, normalized size = 1.47 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 81, normalized size = 2.53 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right )}{a^{2}}+\frac {i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{a^{2}}-\frac {i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 30, normalized size = 0.94 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{a^{2}} + \frac {\arcsin \left (a x\right )}{a^{2}} \]
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 \[ \int \frac {x\,\mathrm {atanh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x \operatorname {atanh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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