Optimal. Leaf size=58 \[ -\frac {1}{a c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}-\frac {\text {Si}(2 \text {ArcTan}(a x))}{a c^3}-\frac {\text {Si}(4 \text {ArcTan}(a x))}{2 a c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5022, 5090,
4491, 3380} \begin {gather*} -\frac {1}{a c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)}-\frac {\text {Si}(2 \text {ArcTan}(a x))}{a c^3}-\frac {\text {Si}(4 \text {ArcTan}(a x))}{2 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4491
Rule 5022
Rule 5090
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac {1}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-(4 a) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx\\ &=-\frac {1}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {4 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=-\frac {1}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {4 \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 c^3}\\ &=-\frac {1}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^3}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=-\frac {1}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{a c^3}-\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 a c^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 45, normalized size = 0.78 \begin {gather*} -\frac {\frac {1}{\left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}+\text {Si}(2 \text {ArcTan}(a x))+\frac {1}{2} \text {Si}(4 \text {ArcTan}(a x))}{a c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 59, normalized size = 1.02
method | result | size |
derivativedivides | \(-\frac {8 \sinIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \sinIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )+4 \cos \left (2 \arctan \left (a x \right )\right )+3}{8 a \,c^{3} \arctan \left (a x \right )}\) | \(59\) |
default | \(-\frac {8 \sinIntegral \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \sinIntegral \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )+4 \cos \left (2 \arctan \left (a x \right )\right )+3}{8 a \,c^{3} \arctan \left (a x \right )}\) | \(59\) |
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 = 2.04, size = 287, normalized size = 4.95 \begin {gather*} \frac {{\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\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 (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\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 ) - 2 \, {\left (i \, a^{4} x^{4} + 2 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 (-i \, a^{4} x^{4} - 2 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 ) - 4}{4 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a 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 {1}{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 {1}{{\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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