Optimal. Leaf size=49 \[ -\frac {2 i \text {ArcTan}(a x)^{5/2}}{5 c}+\frac {i \text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{x (i+a x)},x\right )}{c} \]
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Rubi [A]
time = 0.08, 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 {\text {ArcTan}(a x)^{3/2}}{x \left (c+a^2 c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{x \left (c+a^2 c x^2\right )} \, dx &=-\frac {2 i \tan ^{-1}(a x)^{5/2}}{5 c}+\frac {i \int \frac {\tan ^{-1}(a x)^{3/2}}{x (i+a x)} \, dx}{c}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcTan}(a x)^{3/2}}{x \left (c+a^2 c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x \left (a^{2} c \,x^{2}+c \right )}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} \frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{2} x^{3} + x}\, dx}{c} \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 = -0.02 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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