Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)^2},x\right ) \]
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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 = {}
\begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)^2} \, dx \end {gather*}
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 )^{3/2}}{\tan ^{-1}(a x)^2} \, dx &=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 2.77, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.30, size = 0, normalized size = 0.00 \[\int \frac {x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arctan \left (a x \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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