Optimal. Leaf size=43 \[ -\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}+\frac {\text {Si}(2 \text {ArcTan}(a x))}{a^3 c^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5062, 5090,
4491, 12, 3380} \begin {gather*} \frac {\text {Si}(2 \text {ArcTan}(a x))}{a^3 c^2}-\frac {x^2}{a c^2 \left (a^2 x^2+1\right ) \text {ArcTan}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 4491
Rule 5062
Rule 5090
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a}\\ &=-\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac {x^2}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac {x^2}{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^3 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.93 \begin {gather*} \frac {-\frac {a^2 x^2}{\left (1+a^2 x^2\right ) \text {ArcTan}(a x)}+\text {Si}(2 \text {ArcTan}(a x))}{a^3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 37, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {2 \sinIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )-1}{2 a^{3} c^{2} \arctan \left (a x \right )}\) | \(37\) |
default | \(\frac {2 \sinIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )-1}{2 a^{3} c^{2} \arctan \left (a x \right )}\) | \(37\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.96, size = 123, normalized size = 2.86 \begin {gather*} -\frac {2 \, a^{2} x^{2} - {\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 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )} \arctan \left (a x\right )} \end {gather*}
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 \begin {gather*} \frac {\int \frac {x^{2}}{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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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