Optimal. Leaf size=48 \[ \frac {b x^6}{18 c}+\frac {1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-c^2 x^6\right )}{18 c^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 48, 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 = {6037, 272, 45}
\begin {gather*} \frac {1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac {b x^6}{18 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 6037
Rubi steps
\begin {align*} \int x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{9} x^9 \left (a+b \tanh ^{-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 \tanh ^{-1}\left (c x^3\right )\right )-\frac {1}{18} (b c) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^6\right )\\ &=\frac {1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {1}{18} (b c) \text {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 \tanh ^{-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 = 0.02, size = 53, normalized size = 1.10 \begin {gather*} \frac {b x^6}{18 c}+\frac {a x^9}{9}+\frac {1}{9} b x^9 \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{18 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 45, normalized size = 0.94
method | result | size |
default | \(\frac {x^{9} a}{9}+\frac {b \,x^{9} \arctanh \left (c \,x^{3}\right )}{9}+\frac {b \,x^{6}}{18 c}+\frac {b \ln \left (c^{2} x^{6}-1\right )}{18 c^{3}}\) | \(45\) |
risch | \(\frac {x^{9} b \ln \left (c \,x^{3}+1\right )}{18}-\frac {x^{9} b \ln \left (-c \,x^{3}+1\right )}{18}+\frac {x^{9} a}{9}+\frac {b \,x^{6}}{18 c}+\frac {b \ln \left (c^{2} x^{6}-1\right )}{18 c^{3}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 46, normalized size = 0.96 \begin {gather*} \frac {1}{9} \, a x^{9} + \frac {1}{18} \, {\left (2 \, x^{9} \operatorname {artanh}\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 62, normalized size = 1.29 \begin {gather*} \frac {b c^{3} x^{9} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a c^{3} x^{9} + b c^{2} x^{6} + b \log \left (c^{2} x^{6} - 1\right )}{18 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 57, normalized size = 1.19 \begin {gather*} \frac {1}{18} \, b x^{9} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + \frac {1}{9} \, a x^{9} + \frac {b x^{6}}{18 \, c} + \frac {b \log \left (c^{2} x^{6} - 1\right )}{18 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 61, normalized size = 1.27 \begin {gather*} \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\,\ln \left (c\,x^3+1\right )}{18}-\frac {b\,x^9\,\ln \left (1-c\,x^3\right )}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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