Optimal. Leaf size=113 \[ -\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2}-\frac {2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}+\frac {3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}-\frac {\text {Si}(2 \text {ArcTan}(a x))}{2 a^2 c^3}-\frac {\text {Si}(4 \text {ArcTan}(a x))}{a^2 c^3} \]
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Rubi [A]
time = 0.32, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5088, 5084,
5022, 5090, 4491, 12, 3380} \begin {gather*} -\frac {\text {Si}(2 \text {ArcTan}(a x))}{2 a^2 c^3}-\frac {\text {Si}(4 \text {ArcTan}(a x))}{a^2 c^3}-\frac {x}{2 a c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)^2}+\frac {3}{2 a^2 c^3 \left (a^2 x^2+1\right ) \text {ArcTan}(a x)}-\frac {2}{a^2 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 12
Rule 3380
Rule 4491
Rule 5022
Rule 5084
Rule 5088
Rule 5090
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac {1}{2} (3 a) \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx\\ &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-2 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{2 a c}\\ &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-6 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac {2 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}+\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac {6 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^3}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac {6 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{4 a^2 c^3}-\frac {3 \text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^3}\\ &=-\frac {x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{a^2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 98, normalized size = 0.87 \begin {gather*} -\frac {a x+\text {ArcTan}(a x)-3 a^2 x^2 \text {ArcTan}(a x)+\left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2 \text {Si}(2 \text {ArcTan}(a x))+2 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2 \text {Si}(4 \text {ArcTan}(a x))}{2 a^2 c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 88, normalized size = 0.78
method | result | size |
derivativedivides | \(-\frac {16 \sinIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+8 \sinIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+4 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sin \left (4 \arctan \left (a x \right )\right )+2 \sin \left (2 \arctan \left (a x \right )\right )}{16 a^{2} c^{3} \arctan \left (a x \right )^{2}}\) | \(88\) |
default | \(-\frac {16 \sinIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+8 \sinIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+4 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sin \left (4 \arctan \left (a x \right )\right )+2 \sin \left (2 \arctan \left (a x \right )\right )}{16 a^{2} c^{3} \arctan \left (a x \right )^{2}}\) | \(88\) |
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 = 5.84, size = 318, normalized size = 2.81 \begin {gather*} -\frac {2 \, {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\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 (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\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 (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right )^{2} \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 (3 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{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 )^{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}{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}{{\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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