Optimal. Leaf size=96 \[ -\frac {3 \tan ^{-1}(a x)^2}{4 a^5 c^2}+\frac {x \tan ^{-1}(a x)}{a^4 c^2}+\frac {1}{4 a^5 c^2 \left (a^2 x^2+1\right )}-\frac {\log \left (a^2 x^2+1\right )}{2 a^5 c^2}+\frac {x \tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.18, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4964, 4916, 4846, 260, 4884, 4934} \[ \frac {1}{4 a^5 c^2 \left (a^2 x^2+1\right )}-\frac {\log \left (a^2 x^2+1\right )}{2 a^5 c^2}+\frac {x \tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac {3 \tan ^{-1}(a x)^2}{4 a^5 c^2}+\frac {x \tan ^{-1}(a x)}{a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4884
Rule 4916
Rule 4934
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^4 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {\int \frac {x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2}+\frac {\int \frac {x^2 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2 c}\\ &=\frac {1}{4 a^5 c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}+\frac {\int \tan ^{-1}(a x) \, dx}{a^4 c^2}-\frac {\int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{2 a^4 c}-\frac {\int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^4 c}\\ &=\frac {1}{4 a^5 c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)}{a^4 c^2}+\frac {x \tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^2}{4 a^5 c^2}-\frac {\int \frac {x}{1+a^2 x^2} \, dx}{a^3 c^2}\\ &=\frac {1}{4 a^5 c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)}{a^4 c^2}+\frac {x \tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^2}{4 a^5 c^2}-\frac {\log \left (1+a^2 x^2\right )}{2 a^5 c^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 79, normalized size = 0.82 \[ \frac {\left (4 a^3 x^3+6 a x\right ) \tan ^{-1}(a x)-2 \left (a^2 x^2+1\right ) \log \left (a^2 x^2+1\right )-3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+1}{4 a^5 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 81, normalized size = 0.84 \[ -\frac {3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) + 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) - 1}{4 \, {\left (a^{7} c^{2} x^{2} + a^{5} c^{2}\right )}} \]
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 = 89, normalized size = 0.93 \[ \frac {1}{4 a^{5} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right )}{a^{4} c^{2}}+\frac {x \arctan \left (a x \right )}{2 a^{4} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{2}}{4 a^{5} c^{2}}-\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a^{5} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 114, normalized size = 1.19 \[ \frac {1}{2} \, {\left (\frac {x}{a^{6} c^{2} x^{2} + a^{4} c^{2}} + \frac {2 \, x}{a^{4} c^{2}} - \frac {3 \, \arctan \left (a x\right )}{a^{5} c^{2}}\right )} \arctan \left (a x\right ) + \frac {{\left (3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) + 1\right )} a}{4 \, {\left (a^{8} c^{2} x^{2} + a^{6} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 94, normalized size = 0.98 \[ \frac {1}{2\,a^2\,\left (2\,a^5\,c^2\,x^2+2\,a^3\,c^2\right )}-\frac {\ln \left (a^2\,x^2+1\right )}{2\,a^5\,c^2}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3\,x}{2\,a^6\,c^2}+\frac {x^3}{a^4\,c^2}\right )}{\frac {1}{a^2}+x^2}-\frac {3\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,a^5\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.09, size = 264, normalized size = 2.75 \[ \begin {cases} \frac {4 a^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} - \frac {2 a^{2} x^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} - \frac {3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} + \frac {6 a x \operatorname {atan}{\left (a x \right )}}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} - \frac {2 \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} - \frac {3 \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} + \frac {1}{4 a^{7} c^{2} x^{2} + 4 a^{5} c^{2}} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (\frac {x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {x^{4}}{20 a} + \frac {x^{2}}{10 a^{3}} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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