Optimal. Leaf size=37 \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-c^2 x^6\right )}{6 c} \]
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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6037, 266}
\begin {gather*} \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-c^2 x^6\right )}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 6037
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-(b c) \int \frac {x^5}{1-c^2 x^6} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-c^2 x^6\right )}{6 c}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 42, normalized size = 1.14 \begin {gather*} \frac {a x^3}{3}+\frac {1}{3} b x^3 \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 39, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {a c \,x^{3}+b c \,x^{3} \arctanh \left (c \,x^{3}\right )+\frac {b \ln \left (-c^{2} x^{6}+1\right )}{2}}{3 c}\) | \(39\) |
default | \(\frac {a c \,x^{3}+b c \,x^{3} \arctanh \left (c \,x^{3}\right )+\frac {b \ln \left (-c^{2} x^{6}+1\right )}{2}}{3 c}\) | \(39\) |
risch | \(\frac {x^{3} b \ln \left (c \,x^{3}+1\right )}{6}-\frac {b \,x^{3} \ln \left (-c \,x^{3}+1\right )}{6}+\frac {x^{3} a}{3}+\frac {b \ln \left (c^{2} x^{6}-1\right )}{6 c}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 37, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, a x^{3} + \frac {{\left (2 \, c x^{3} \operatorname {artanh}\left (c x^{3}\right ) + \log \left (-c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 50, normalized size = 1.35 \begin {gather*} \frac {b c x^{3} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a c x^{3} + b \log \left (c^{2} x^{6} - 1\right )}{6 \, c} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs.
\(2 (33) = 66\).
time = 0.44, size = 188, normalized size = 5.08 \begin {gather*} \frac {1}{3} \, a x^{3} + \frac {1}{3} \, b c {\left (\frac {\log \left (\frac {{\left | -c x^{3} - 1 \right |}}{{\left | c x^{3} - 1 \right |}}\right )}{c^{2}} - \frac {\log \left ({\left | -\frac {c x^{3} + 1}{c x^{3} - 1} + 1 \right |}\right )}{c^{2}} + \frac {\log \left (-\frac {\frac {c {\left (\frac {c x^{3} + 1}{c x^{3} - 1} + 1\right )}}{\frac {{\left (c x^{3} + 1\right )} c}{c x^{3} - 1} - c} + 1}{\frac {c {\left (\frac {c x^{3} + 1}{c x^{3} - 1} + 1\right )}}{\frac {{\left (c x^{3} + 1\right )} c}{c x^{3} - 1} - c} - 1}\right )}{c^{2} {\left (\frac {c x^{3} + 1}{c x^{3} - 1} - 1\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 52, normalized size = 1.41 \begin {gather*} \frac {a\,x^3}{3}+\frac {b\,\ln \left (c^2\,x^6-1\right )}{6\,c}+\frac {b\,x^3\,\ln \left (c\,x^3+1\right )}{6}-\frac {b\,x^3\,\ln \left (1-c\,x^3\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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