Optimal. Leaf size=61 \[ -\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}+\frac {\text {CosIntegral}(2 \text {ArcTan}(a x))}{2 a^2 c^3}+\frac {\text {CosIntegral}(4 \text {ArcTan}(a x))}{2 a^2 c^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5088, 5090,
4491, 3383, 5024, 3393} \begin {gather*} \frac {\text {CosIntegral}(2 \text {ArcTan}(a x))}{2 a^2 c^3}+\frac {\text {CosIntegral}(4 \text {ArcTan}(a x))}{2 a^2 c^3}-\frac {x}{a c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 4491
Rule 5024
Rule 5088
Rule 5090
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a}-(3 a) \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx\\ &=-\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=-\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac {3 \text {Subst}\left (\int \left (\frac {1}{8 x}-\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=-\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^3}\\ &=-\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}+\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{2 a^2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 75, normalized size = 1.23 \begin {gather*} \frac {-2 a x+\left (1+a^2 x^2\right )^2 \text {ArcTan}(a x) \text {CosIntegral}(2 \text {ArcTan}(a x))+\left (1+a^2 x^2\right )^2 \text {ArcTan}(a x) \text {CosIntegral}(4 \text {ArcTan}(a x))}{2 c^3 \left (a+a^3 x^2\right )^2 \text {ArcTan}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 60, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {4 \cosineIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \cosineIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (4 \arctan \left (a x \right )\right )-2 \sin \left (2 \arctan \left (a x \right )\right )}{8 a^{2} c^{3} \arctan \left (a x \right )}\) | \(60\) |
default | \(\frac {4 \cosineIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \cosineIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\sin \left (4 \arctan \left (a x \right )\right )-2 \sin \left (2 \arctan \left (a x \right )\right )}{8 a^{2} c^{3} \arctan \left (a x \right )}\) | \(60\) |
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 = 1.13, size = 286, normalized size = 4.69 \begin {gather*} \frac {{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, 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^{4} x^{4} + 2 \, 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 ) - 4 \, a x}{4 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\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}{a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \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}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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