Optimal. Leaf size=112 \[ -\frac {4 \text {Int}\left (\frac {1}{x^5 \tan ^{-1}(a x)},x\right )}{a c^2}+\frac {2 a \text {Int}\left (\frac {1}{x^3 \tan ^{-1}(a x)},x\right )}{c^2}-\frac {a^3 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {a^3}{c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.40, 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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx\right )+\frac {\int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+a^4 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx-\frac {4 \int \frac {1}{x^5 \tan ^{-1}(a x)} \, dx}{a c^2}-\frac {a^2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)}-\frac {a^3}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\left (2 a^5\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx-\frac {4 \int \frac {1}{x^5 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac {(2 a) \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{c^2}\\ &=-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)}-\frac {a^3}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {4 \int \frac {1}{x^5 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac {(2 a) \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{c^2}-\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)}-\frac {a^3}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {4 \int \frac {1}{x^5 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac {(2 a) \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{c^2}-\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)}-\frac {a^3}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {4 \int \frac {1}{x^5 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac {(2 a) \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{c^2}-\frac {a^3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)}-\frac {a^3}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a^3 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {4 \int \frac {1}{x^5 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac {(2 a) \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{c^2}\\ \end {align*}
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Mathematica [A] time = 3.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{8} + 2 \, a^{2} c^{2} x^{6} + c^{2} x^{4}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]
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 = 2.49, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}\, dx \]
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 \[ -\frac {2 \, {\left (a^{3} c^{2} x^{6} + a c^{2} x^{4}\right )} \arctan \left (a x\right ) \int \frac {3 \, a^{2} x^{2} + 2}{{\left (a^{5} c^{2} x^{9} + 2 \, a^{3} c^{2} x^{7} + a c^{2} x^{5}\right )} \arctan \left (a x\right )}\,{d x} + 1}{{\left (a^{3} c^{2} x^{6} + a c^{2} x^{4}\right )} \arctan \left (a x\right )} \]
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 \[ \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {1}{a^{4} x^{8} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + x^{4} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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