Optimal. Leaf size=41 \[ -\frac {1}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2} \]
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Rubi [A] time = 0.09, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4902, 4970, 4406, 12, 3299} \[ -\frac {1}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 4902
Rule 4970
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac {1}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-(2 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx\\ &=-\frac {1}{a 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 )}{a c^2}\\ &=-\frac {1}{a 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 )}{a c^2}\\ &=-\frac {1}{a 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 )}{a c^2}\\ &=-\frac {1}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 34, normalized size = 0.83 \[ -\frac {\frac {1}{a^2 x^2 \tan ^{-1}(a x)+\tan ^{-1}(a x)}+\text {Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.51, size = 112, normalized size = 2.73 \[ \frac {{\left (-i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2}{2 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] 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 = 0.23, size = 37, normalized size = 0.90 \[ -\frac {2 \Si \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )+1}{2 a \,c^{2} \arctan \left (a x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right ) \int \frac {x}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )}\,{d x} + 1}{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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