Optimal. Leaf size=50 \[ -\frac {\text {CosIntegral}(2 \text {ArcTan}(a x))}{2 a^5 c^3}+\frac {\text {CosIntegral}(4 \text {ArcTan}(a x))}{8 a^5 c^3}+\frac {3 \log (\text {ArcTan}(a x))}{8 a^5 c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5090, 3393,
3383} \begin {gather*} -\frac {\text {CosIntegral}(2 \text {ArcTan}(a x))}{2 a^5 c^3}+\frac {\text {CosIntegral}(4 \text {ArcTan}(a x))}{8 a^5 c^3}+\frac {3 \log (\text {ArcTan}(a x))}{8 a^5 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 5090
Rubi steps
\begin {align*} \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^3}\\ &=\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^5 c^3}\\ &=\frac {3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^5 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^5 c^3}\\ &=-\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^5 c^3}+\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{8 a^5 c^3}+\frac {3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 34, normalized size = 0.68 \begin {gather*} \frac {-4 \text {CosIntegral}(2 \text {ArcTan}(a x))+\text {CosIntegral}(4 \text {ArcTan}(a x))+3 \log (\text {ArcTan}(a x))}{8 a^5 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.12, size = 40, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {3 \ln \left (\arctan \left (a x \right )\right )}{8 c^{3}}-\frac {\cosineIntegral \left (2 \arctan \left (a x \right )\right )}{2 c^{3}}+\frac {\cosineIntegral \left (4 \arctan \left (a x \right )\right )}{8 c^{3}}}{a^{5}}\) | \(40\) |
default | \(\frac {\frac {3 \ln \left (\arctan \left (a x \right )\right )}{8 c^{3}}-\frac {\cosineIntegral \left (2 \arctan \left (a x \right )\right )}{2 c^{3}}+\frac {\cosineIntegral \left (4 \arctan \left (a x \right )\right )}{8 c^{3}}}{a^{5}}\) | \(40\) |
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 = 1.32, size = 174, normalized size = 3.48 \begin {gather*} \frac {6 \, \log \left (\arctan \left (a x\right )\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 ) + \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 ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{16 \, a^{5} c^{3}} \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^{4}}{a^{6} x^{6} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\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^4}{\mathrm {atan}\left (a\,x\right )\,{\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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