Optimal. Leaf size=153 \[ \frac {c^2 \left (a^2 x^2+1\right )^2}{60 a^2}+\frac {2 c^2 \left (a^2 x^2+1\right )}{45 a^2}+\frac {4 c^2 \log \left (a^2 x^2+1\right )}{45 a^2}+\frac {c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac {c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{15 a}-\frac {4 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{45 a}-\frac {8 c^2 x \tan ^{-1}(a x)}{45 a} \]
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Rubi [A] time = 0.09, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4930, 4878, 4846, 260} \[ \frac {c^2 \left (a^2 x^2+1\right )^2}{60 a^2}+\frac {2 c^2 \left (a^2 x^2+1\right )}{45 a^2}+\frac {4 c^2 \log \left (a^2 x^2+1\right )}{45 a^2}+\frac {c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac {c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{15 a}-\frac {4 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{45 a}-\frac {8 c^2 x \tan ^{-1}(a x)}{45 a} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4878
Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac {\int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx}{3 a}\\ &=\frac {c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac {(4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx}{15 a}\\ &=\frac {2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac {4 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{45 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac {\left (8 c^2\right ) \int \tan ^{-1}(a x) \, dx}{45 a}\\ &=\frac {2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac {8 c^2 x \tan ^{-1}(a x)}{45 a}-\frac {4 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{45 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}+\frac {1}{45} \left (8 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=\frac {2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac {8 c^2 x \tan ^{-1}(a x)}{45 a}-\frac {4 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{45 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}+\frac {4 c^2 \log \left (1+a^2 x^2\right )}{45 a^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.55 \[ \frac {c^2 \left (3 a^4 x^4+14 a^2 x^2+16 \log \left (a^2 x^2+1\right )+30 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2-4 a x \left (3 a^4 x^4+10 a^2 x^2+15\right ) \tan ^{-1}(a x)\right )}{180 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 123, normalized size = 0.80 \[ \frac {3 \, a^{4} c^{2} x^{4} + 14 \, a^{2} c^{2} x^{2} + 30 \, {\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arctan \left (a x\right )}{180 \, a^{2}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 142, normalized size = 0.93 \[ \frac {a^{4} c^{2} \arctan \left (a x \right )^{2} x^{6}}{6}+\frac {a^{2} c^{2} \arctan \left (a x \right )^{2} x^{4}}{2}+\frac {c^{2} \arctan \left (a x \right )^{2} x^{2}}{2}-\frac {a^{3} c^{2} \arctan \left (a x \right ) x^{5}}{15}-\frac {2 a \,c^{2} \arctan \left (a x \right ) x^{3}}{9}-\frac {c^{2} x \arctan \left (a x \right )}{3 a}+\frac {c^{2} \arctan \left (a x \right )^{2}}{6 a^{2}}+\frac {a^{2} c^{2} x^{4}}{60}+\frac {7 c^{2} x^{2}}{90}+\frac {4 c^{2} \ln \left (a^{2} x^{2}+1\right )}{45 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 111, normalized size = 0.73 \[ \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{6 \, a^{2} c} + \frac {{\left (3 \, a^{2} c^{3} x^{4} + 14 \, c^{3} x^{2} + \frac {16 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 4 \, {\left (3 \, a^{4} c^{3} x^{5} + 10 \, a^{2} c^{3} x^{3} + 15 \, c^{3} x\right )} \arctan \left (a x\right )}{180 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 135, normalized size = 0.88 \[ \frac {\frac {c^2\,\left (30\,{\mathrm {atan}\left (a\,x\right )}^2+16\,\ln \left (a^2\,x^2+1\right )\right )}{180}-\frac {a\,c^2\,x\,\mathrm {atan}\left (a\,x\right )}{3}}{a^2}+\frac {c^2\,\left (90\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+14\,x^2\right )}{180}+\frac {a^2\,c^2\,\left (90\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2+3\,x^4\right )}{180}-\frac {a^3\,c^2\,x^5\,\mathrm {atan}\left (a\,x\right )}{15}+\frac {a^4\,c^2\,x^6\,{\mathrm {atan}\left (a\,x\right )}^2}{6}-\frac {2\,a\,c^2\,x^3\,\mathrm {atan}\left (a\,x\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.02, size = 158, normalized size = 1.03 \[ \begin {cases} \frac {a^{4} c^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{6} - \frac {a^{3} c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{15} + \frac {a^{2} c^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {a^{2} c^{2} x^{4}}{60} - \frac {2 a c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{9} + \frac {c^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {7 c^{2} x^{2}}{90} - \frac {c^{2} x \operatorname {atan}{\left (a x \right )}}{3 a} + \frac {4 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{45 a^{2}} + \frac {c^{2} \operatorname {atan}^{2}{\left (a x \right )}}{6 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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