Optimal. Leaf size=114 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tan ^{-1}(a x)^2},x\right )}{2 a c^2}+\frac {a x}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2} \]
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Rubi [A] time = 0.26, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx\right )+\frac {\int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c}\\ &=-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\left (2 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ \end {align*}
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Mathematica [A] time = 1.94, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \arctan \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}\, dx \]
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 \[ \frac {2 \, {\left (a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )} \mathit {sage}_{0} x \arctan \left (a x\right )^{2} - a x + {\left (3 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )} + x \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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