Optimal. Leaf size=88 \[ -\frac {4 a^3 \log (x)}{3 c}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}+\frac {a^2 \tan ^{-1}(a x)}{c x}+\frac {2 a^3 \log \left (a^2 x^2+1\right )}{3 c}-\frac {\tan ^{-1}(a x)}{3 c x^3}-\frac {a}{6 c x^2} \]
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Rubi [A] time = 0.16, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4918, 4852, 266, 44, 36, 29, 31, 4884} \[ \frac {2 a^3 \log \left (a^2 x^2+1\right )}{3 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}+\frac {a^2 \tan ^{-1}(a x)}{c x}-\frac {a}{6 c x^2}-\frac {\tan ^{-1}(a x)}{3 c x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 4852
Rule 4884
Rule 4918
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^4} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)}{3 c x^3}+a^4 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx+\frac {a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)}{c x}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )}{6 c}-\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)}{c x}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}+\frac {a \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )}{6 c}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {a}{6 c x^2}-\frac {\tan ^{-1}(a x)}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)}{c x}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}-\frac {a^3 \log (x)}{3 c}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}+\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {a}{6 c x^2}-\frac {\tan ^{-1}(a x)}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)}{c x}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 88, normalized size = 1.00 \[ -\frac {4 a^3 \log (x)}{3 c}+\frac {a^3 \tan ^{-1}(a x)^2}{2 c}+\frac {a^2 \tan ^{-1}(a x)}{c x}+\frac {2 a^3 \log \left (a^2 x^2+1\right )}{3 c}-\frac {\tan ^{-1}(a x)}{3 c x^3}-\frac {a}{6 c x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 71, normalized size = 0.81 \[ \frac {3 \, a^{3} x^{3} \arctan \left (a x\right )^{2} + 4 \, a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{3} x^{3} \log \relax (x) - a x + 2 \, {\left (3 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{6 \, c x^{3}} \]
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 = 0.04, size = 81, normalized size = 0.92 \[ -\frac {\arctan \left (a x \right )}{3 c \,x^{3}}+\frac {a^{2} \arctan \left (a x \right )}{c x}+\frac {a^{3} \arctan \left (a x \right )^{2}}{2 c}-\frac {a}{6 c \,x^{2}}-\frac {4 a^{3} \ln \left (a x \right )}{3 c}+\frac {2 a^{3} \ln \left (a^{2} x^{2}+1\right )}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 90, normalized size = 1.02 \[ \frac {1}{3} \, {\left (\frac {3 \, a^{3} \arctan \left (a x\right )}{c} + \frac {3 \, a^{2} x^{2} - 1}{c x^{3}}\right )} \arctan \left (a x\right ) - \frac {{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \relax (x) + 1\right )} a}{6 \, c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 78, normalized size = 0.89 \[ \frac {2\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c}-\frac {\mathrm {atan}\left (a\,x\right )}{3\,c\,x^3}-\frac {a}{6\,c\,x^2}-\frac {4\,a^3\,\ln \relax (x)}{3\,c}+\frac {a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,c}+\frac {a^2\,\mathrm {atan}\left (a\,x\right )}{c\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.53, size = 117, normalized size = 1.33 \[ \begin {cases} - \frac {4 a^{3} \log {\relax (x )}}{3 c} + \frac {2 a^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 c} + \frac {a^{3} \operatorname {atan}^{2}{\left (a x \right )}}{2 c} + \frac {a^{2} \operatorname {atan}{\left (a x \right )}}{c x} - \frac {a}{6 c x^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{3 c x^{3}} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (- \frac {a^{3} \log {\relax (x )}}{3} + \frac {a^{3} \log {\left (a^{2} x^{2} + 1 \right )}}{6} - \frac {a}{6 x^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{3 x^{3}}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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