Optimal. Leaf size=37 \[ \frac {2 \text {Int}\left (\frac {1}{\sqrt {\tan ^{-1}(a x)}},x\right )}{a c}-\frac {2 x}{a c \sqrt {\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x}{a c \sqrt {\tan ^{-1}(a x)}}+\frac {2 \int \frac {1}{\sqrt {\tan ^{-1}(a x)}} \, dx}{a c}\\ \end {align*}
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Mathematica [A] time = 1.13, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x}{a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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