Optimal. Leaf size=55 \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{9 x^9}-\frac {1}{3} b c^3 \log (x)+\frac {1}{18} b c^3 \log \left (c^2 x^6+1\right )-\frac {b c}{18 x^6} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5033, 266, 44} \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{9 x^9}+\frac {1}{18} b c^3 \log \left (c^2 x^6+1\right )-\frac {1}{3} b c^3 \log (x)-\frac {b c}{18 x^6} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 5033
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}\left (c x^3\right )}{x^{10}} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{9 x^9}+\frac {1}{3} (b c) \int \frac {1}{x^7 \left (1+c^2 x^6\right )} \, dx\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{9 x^9}+\frac {1}{18} (b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^6\right )\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{9 x^9}+\frac {1}{18} (b c) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^6\right )\\ &=-\frac {b c}{18 x^6}-\frac {a+b \tan ^{-1}\left (c x^3\right )}{9 x^9}-\frac {1}{3} b c^3 \log (x)+\frac {1}{18} b c^3 \log \left (1+c^2 x^6\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 1.09 \[ -\frac {a}{9 x^9}-\frac {1}{3} b c^3 \log (x)+\frac {1}{18} b c^3 \log \left (c^2 x^6+1\right )-\frac {b c}{18 x^6}-\frac {b \tan ^{-1}\left (c x^3\right )}{9 x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 54, normalized size = 0.98 \[ \frac {b c^{3} x^{9} \log \left (c^{2} x^{6} + 1\right ) - 6 \, b c^{3} x^{9} \log \relax (x) - b c x^{3} - 2 \, b \arctan \left (c x^{3}\right ) - 2 \, a}{18 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 69, normalized size = 1.25 \[ \frac {b c^{7} x^{9} \log \left (c^{2} x^{6} + 1\right ) - 2 \, b c^{7} x^{9} \log \left (c x^{3}\right ) - b c^{5} x^{3} - 2 \, b c^{4} \arctan \left (c x^{3}\right ) - 2 \, a c^{4}}{18 \, c^{4} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 51, normalized size = 0.93 \[ -\frac {a}{9 x^{9}}-\frac {b \arctan \left (c \,x^{3}\right )}{9 x^{9}}+\frac {b \,c^{3} \ln \left (c^{2} x^{6}+1\right )}{18}-\frac {b c}{18 x^{6}}-\frac {b \,c^{3} \ln \relax (x )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 53, normalized size = 0.96 \[ \frac {1}{18} \, {\left ({\left (c^{2} \log \left (c^{2} x^{6} + 1\right ) - c^{2} \log \left (x^{6}\right ) - \frac {1}{x^{6}}\right )} c - \frac {2 \, \arctan \left (c x^{3}\right )}{x^{9}}\right )} b - \frac {a}{9 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 50, normalized size = 0.91 \[ \frac {b\,c^3\,\ln \left (c^2\,x^6+1\right )}{18}-\frac {a}{9\,x^9}-\frac {b\,c^3\,\ln \relax (x)}{3}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{9\,x^9}-\frac {b\,c}{18\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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