Optimal. Leaf size=100 \[ -\frac {2}{a c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}}-\frac {6 \text {Int}\left (\frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {ArcTan}(a x)}},x\right )}{a}-12 a \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {ArcTan}(a x)}},x\right ) \]
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Rubi [A]
time = 0.23, 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 {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {6 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(12 a) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}
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Mathematica [A]
time = 19.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 2.31, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]
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 \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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