Optimal. Leaf size=117 \[ -\frac {2 c^2 \left (1+a^2 x^2\right )}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2}{20 a}+\frac {8}{15} c^2 x \text {ArcTan}(a x)+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \text {ArcTan}(a x)+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)-\frac {4 c^2 \log \left (1+a^2 x^2\right )}{15 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4998, 4930,
266} \begin {gather*} \frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)+\frac {4}{15} c^2 x \left (a^2 x^2+1\right ) \text {ArcTan}(a x)-\frac {c^2 \left (a^2 x^2+1\right )^2}{20 a}-\frac {2 c^2 \left (a^2 x^2+1\right )}{15 a}-\frac {4 c^2 \log \left (a^2 x^2+1\right )}{15 a}+\frac {8}{15} c^2 x \text {ArcTan}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4930
Rule 4998
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx &=-\frac {c^2 \left (1+a^2 x^2\right )^2}{20 a}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac {1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx\\ &=-\frac {2 c^2 \left (1+a^2 x^2\right )}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2}{20 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac {1}{15} \left (8 c^2\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac {2 c^2 \left (1+a^2 x^2\right )}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2}{20 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac {1}{15} \left (8 a c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-\frac {2 c^2 \left (1+a^2 x^2\right )}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2}{20 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac {4 c^2 \log \left (1+a^2 x^2\right )}{15 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 65, normalized size = 0.56 \begin {gather*} \frac {c^2 \left (-14 a^2 x^2-3 a^4 x^4+4 a x \left (15+10 a^2 x^2+3 a^4 x^4\right ) \text {ArcTan}(a x)-16 \log \left (1+a^2 x^2\right )\right )}{60 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 80, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 a^{3} c^{2} x^{3} \arctan \left (a x \right )}{3}+a \,c^{2} x \arctan \left (a x \right )-\frac {c^{2} \left (\frac {3 a^{4} x^{4}}{4}+\frac {7 a^{2} x^{2}}{2}+4 \ln \left (a^{2} x^{2}+1\right )\right )}{15}}{a}\) | \(80\) |
default | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 a^{3} c^{2} x^{3} \arctan \left (a x \right )}{3}+a \,c^{2} x \arctan \left (a x \right )-\frac {c^{2} \left (\frac {3 a^{4} x^{4}}{4}+\frac {7 a^{2} x^{2}}{2}+4 \ln \left (a^{2} x^{2}+1\right )\right )}{15}}{a}\) | \(80\) |
risch | \(-\frac {i c^{2} x \left (3 a^{4} x^{4}+10 a^{2} x^{2}+15\right ) \ln \left (i a x +1\right )}{30}+\frac {i c^{2} a^{4} x^{5} \ln \left (-i a x +1\right )}{10}-\frac {c^{2} a^{3} x^{4}}{20}+\frac {i c^{2} a^{2} x^{3} \ln \left (-i a x +1\right )}{3}-\frac {7 a \,c^{2} x^{2}}{30}+\frac {i c^{2} x \ln \left (-i a x +1\right )}{2}-\frac {4 c^{2} \ln \left (-a^{2} x^{2}-1\right )}{15 a}-\frac {49 c^{2}}{180 a}\) | \(137\) |
meijerg | \(\frac {c^{2} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4 a}+\frac {c^{2} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2 a}+\frac {c^{2} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 77, normalized size = 0.66 \begin {gather*} -\frac {1}{60} \, {\left (3 \, a^{2} c^{2} x^{4} + 14 \, c^{2} x^{2} + \frac {16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arctan \left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.31, size = 79, normalized size = 0.68 \begin {gather*} -\frac {3 \, a^{4} c^{2} x^{4} + 14 \, a^{2} c^{2} x^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arctan \left (a x\right )}{60 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 88, normalized size = 0.75 \begin {gather*} \begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {a^{3} c^{2} x^{4}}{20} + \frac {2 a^{2} c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {7 a c^{2} x^{2}}{30} + c^{2} x \operatorname {atan}{\left (a x \right )} - \frac {4 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{15 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \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 [B]
time = 0.20, size = 69, normalized size = 0.59 \begin {gather*} -\frac {c^2\,\left (16\,\ln \left (a^2\,x^2+1\right )+14\,a^2\,x^2+3\,a^4\,x^4-40\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )-12\,a^5\,x^5\,\mathrm {atan}\left (a\,x\right )-60\,a\,x\,\mathrm {atan}\left (a\,x\right )\right )}{60\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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