Optimal. Leaf size=61 \[ \frac {2 a^3 \text {ArcTan}(a x)^{5/2}}{5 c}+\frac {\text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{x^4},x\right )}{c}-\frac {a^2 \text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{x^2},x\right )}{c} \]
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Rubi [A]
time = 0.14, 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^4 \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^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2} \, dx}{c}\\ &=\frac {2 a^3 \tan ^{-1}(a x)^{5/2}}{5 c}+\frac {\int \frac {\tan ^{-1}(a x)^{3/2}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2} \, dx}{c}\\ \end {align*}
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Mathematica [A]
time = 2.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcTan}(a x)^{3/2}}{x^4 \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 = 1.51, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x^{4} \left (a^{2} c \,x^{2}+c \right )}\, dx\]
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^{6} + x^{4}}\, 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^4\,\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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