Optimal. Leaf size=39 \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}-\frac {1}{6} b c \log \left (c^2 x^6+1\right )+b c \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5033, 266, 36, 29, 31} \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}-\frac {1}{6} b c \log \left (c^2 x^6+1\right )+b c \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5033
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}\left (c x^3\right )}{x^4} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}+(b c) \int \frac {1}{x \left (1+c^2 x^6\right )} \, dx\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^6\right )\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^6\right )-\frac {1}{6} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^6\right )\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}+b c \log (x)-\frac {1}{6} b c \log \left (1+c^2 x^6\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.13 \[ -\frac {a}{3 x^3}-\frac {1}{6} b c \log \left (c^2 x^6+1\right )-\frac {b \tan ^{-1}\left (c x^3\right )}{3 x^3}+b c \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 43, normalized size = 1.10 \[ -\frac {b c x^{3} \log \left (c^{2} x^{6} + 1\right ) - 6 \, b c x^{3} \log \relax (x) + 2 \, b \arctan \left (c x^{3}\right ) + 2 \, a}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.01, size = 60, normalized size = 1.54 \[ -\frac {b c^{3} x^{3} \log \left (c^{2} x^{6} + 1\right ) - 2 \, b c^{3} x^{3} \log \left (c x^{3}\right ) + 2 \, b c^{2} \arctan \left (c x^{3}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 39, normalized size = 1.00 \[ -\frac {a}{3 x^{3}}-\frac {b \arctan \left (c \,x^{3}\right )}{3 x^{3}}-\frac {b c \ln \left (c^{2} x^{6}+1\right )}{6}+b c \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 41, normalized size = 1.05 \[ -\frac {1}{6} \, {\left (c {\left (\log \left (c^{2} x^{6} + 1\right ) - \log \left (x^{6}\right )\right )} + \frac {2 \, \arctan \left (c x^{3}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 38, normalized size = 0.97 \[ b\,c\,\ln \relax (x)-\frac {a}{3\,x^3}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^3}-\frac {b\,c\,\ln \left (c^2\,x^6+1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 96.09, size = 126, normalized size = 3.23 \[ \begin {cases} - \frac {a}{3 x^{3}} + b c \log {\relax (x )} - \frac {b c \log {\left (x - \sqrt [6]{-1} \sqrt [6]{\frac {1}{c^{2}}} \right )}}{3} - \frac {b c \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{3} + \frac {i b \operatorname {atan}{\left (c x^{3} \right )}}{3 \sqrt {\frac {1}{c^{2}}}} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{3}} & \text {for}\: c \neq 0 \\- \frac {a}{3 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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