Optimal. Leaf size=47 \[ \frac {1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (c^2 x^6+1\right )}{18 c^3}-\frac {b x^6}{18 c} \]
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Rubi [A] time = 0.04, antiderivative size = 47, 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, 43} \[ \frac {1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (c^2 x^6+1\right )}{18 c^3}-\frac {b x^6}{18 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 5033
Rubi steps
\begin {align*} \int x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{3} (b c) \int \frac {x^{11}}{1+c^2 x^6} \, dx\\ &=\frac {1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{18} (b c) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^6\right )\\ &=\frac {1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{18} (b c) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^6\right )\\ &=-\frac {b x^6}{18 c}+\frac {1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1+c^2 x^6\right )}{18 c^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.11 \[ \frac {a x^9}{9}+\frac {b \log \left (c^2 x^6+1\right )}{18 c^3}-\frac {b x^6}{18 c}+\frac {1}{9} b x^9 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 51, normalized size = 1.09 \[ \frac {2 \, b c^{3} x^{9} \arctan \left (c x^{3}\right ) + 2 \, a c^{3} x^{9} - b c^{2} x^{6} + b \log \left (c^{2} x^{6} + 1\right )}{18 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.81, size = 47, normalized size = 1.00 \[ \frac {2 \, a c x^{9} + {\left (2 \, c x^{9} \arctan \left (c x^{3}\right ) - x^{6} + \frac {\log \left (c^{2} x^{6} + 1\right )}{c^{2}}\right )} b}{18 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 45, normalized size = 0.96 \[ \frac {x^{9} a}{9}+\frac {b \,x^{9} \arctan \left (c \,x^{3}\right )}{9}-\frac {b \,x^{6}}{18 c}+\frac {b \ln \left (c^{2} x^{6}+1\right )}{18 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 48, normalized size = 1.02 \[ \frac {1}{9} \, a x^{9} + \frac {1}{18} \, {\left (2 \, x^{9} \arctan \left (c x^{3}\right ) - {\left (\frac {x^{6}}{c^{2}} - \frac {\log \left (c^{2} x^{6} + 1\right )}{c^{4}}\right )} c\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 44, normalized size = 0.94 \[ \frac {a\,x^9}{9}+\frac {b\,\ln \left (c^2\,x^6+1\right )}{18\,c^3}-\frac {b\,x^6}{18\,c}+\frac {b\,x^9\,\mathrm {atan}\left (c\,x^3\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 177.26, size = 133, normalized size = 2.83 \[ \begin {cases} \frac {a x^{9}}{9} + \frac {b x^{9} \operatorname {atan}{\left (c x^{3} \right )}}{9} - \frac {b x^{6}}{18 c} - \frac {i b \sqrt {\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{9 c^{2}} + \frac {b \log {\left (x - \sqrt [6]{-1} \sqrt [6]{\frac {1}{c^{2}}} \right )}}{9 c^{3}} + \frac {b \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {a x^{9}}{9} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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