Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^2}{x \tan ^{-1}(a x)^3},x\right ) \]
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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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^2}{x \tan ^{-1}(a x)^3} \, dx &=\int \frac {\left (c+a^2 c x^2\right )^2}{x \tan ^{-1}(a x)^3} \, dx\\ \end {align*}
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Mathematica [A] time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}{x \arctan \left (a x\right )^{3}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.88, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{2}}{x \arctan \left (a x \right )^{3}}\, dx \]
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 \[ -\frac {a^{7} c^{2} x^{7} + 3 \, a^{5} c^{2} x^{5} + 3 \, a^{3} c^{2} x^{3} - 2 \, \mathit {sage}_{0} x^{3} \arctan \left (a x\right )^{2} + a c^{2} x + {\left (5 \, a^{8} c^{2} x^{8} + 14 \, a^{6} c^{2} x^{6} + 12 \, a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} - c^{2}\right )} \arctan \left (a x\right )}{2 \, a^{2} x^{2} \arctan \left (a x\right )^{2}} \]
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 \[ \int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{x\,{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]
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 \[ c^{2} \left (\int \frac {1}{x \operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {2 a^{2} x}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{3}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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