Optimal. Leaf size=120 \[ \frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)^2}-\frac {2 x}{a^2 c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}+\frac {x}{a^2 c^3 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}+\frac {\text {CosIntegral}(4 \text {ArcTan}(a x))}{a^3 c^3} \]
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Rubi [A]
time = 0.41, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5084, 5022,
5088, 5090, 3393, 3383, 5024, 4491} \begin {gather*} \frac {\text {CosIntegral}(4 \text {ArcTan}(a x))}{a^3 c^3}+\frac {x}{a^2 c^3 \left (a^2 x^2+1\right ) \text {ArcTan}(a x)}-\frac {2 x}{a^2 c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)}-\frac {1}{2 a^3 c^3 \left (a^2 x^2+1\right ) \text {ArcTan}(a x)^2}+\frac {1}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 4491
Rule 5022
Rule 5024
Rule 5084
Rule 5088
Rule 5090
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx}{a^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{a^2 c}\\ &=\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {2 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{a}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{a c}\\ &=\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac {2 x}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {x}{a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-6 \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a^2}+\frac {\int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^2 c}\\ &=\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac {2 x}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {x}{a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac {2 \text {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}-\frac {6 \text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac {2 x}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {x}{a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac {2 \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^3 c^3}-\frac {6 \text {Subst}\left (\int \left (\frac {1}{8 x}-\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac {2 x}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {x}{a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^3}-2 \frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^3}+\frac {3 \text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^3}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac {2 x}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {x}{a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{a^3 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 60, normalized size = 0.50 \begin {gather*} \frac {\frac {a x \left (-a x+2 \left (-1+a^2 x^2\right ) \text {ArcTan}(a x)\right )}{\left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2}+2 \text {CosIntegral}(4 \text {ArcTan}(a x))}{2 a^3 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 52, normalized size = 0.43
method | result | size |
derivativedivides | \(\frac {16 \cosineIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-4 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )-1}{16 a^{3} c^{3} \arctan \left (a x \right )^{2}}\) | \(52\) |
default | \(\frac {16 \cosineIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-4 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )-1}{16 a^{3} c^{3} \arctan \left (a x \right )^{2}}\) | \(52\) |
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 = 3.25, size = 215, normalized size = 1.79 \begin {gather*} -\frac {a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} \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 )^{2} \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 ) - 2 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{2 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )} \arctan \left (a x\right )^{2}} \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^{6} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\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 = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^3\,{\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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