Optimal. Leaf size=41 \[ \frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.21, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4968, 4970, 3312, 3302, 4904} \[ \frac {\text {CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{a^2 c^2}-\frac {x}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 4904
Rule 4968
Rule 4970
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a}-a \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx\\ &=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}-\frac {\operatorname {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}\\ &=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}\\ &=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+2 \frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^2}\\ &=-\frac {x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{a^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 36, normalized size = 0.88 \[ \frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )-\frac {a x}{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)}}{a^2 c^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 115, normalized size = 2.80 \[ \frac {{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, a x}{2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right )} \]
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 = 0.25, size = 38, normalized size = 0.93 \[ \frac {2 \Ci \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (2 \arctan \left (a x \right )\right )}{2 a^{2} c^{2} \arctan \left (a x \right )} \]
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 \[ -\frac {{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \mathit {sage}_{0} x \arctan \left (a x\right ) + x}{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )} \]
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 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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