Optimal. Leaf size=43 \[ \frac {b x^3}{6 c}-\frac {b \tanh ^{-1}\left (c x^3\right )}{6 c^2}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6037, 281, 327,
212} \begin {gather*} \frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {b \tanh ^{-1}\left (c x^3\right )}{6 c^2}+\frac {b x^3}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rule 327
Rule 6037
Rubi steps
\begin {align*} \int x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {1}{2} (b c) \int \frac {x^8}{1-c^2 x^6} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,x^3\right )\\ &=\frac {b x^3}{6 c}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^3\right )}{6 c}\\ &=\frac {b x^3}{6 c}-\frac {b \tanh ^{-1}\left (c x^3\right )}{6 c^2}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 1.56 \begin {gather*} \frac {b x^3}{6 c}+\frac {a x^6}{6}+\frac {1}{6} b x^6 \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c x^3\right )}{12 c^2}-\frac {b \log \left (1+c x^3\right )}{12 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 57, normalized size = 1.33
method | result | size |
default | \(\frac {x^{6} a}{6}+\frac {b \,x^{6} \arctanh \left (c \,x^{3}\right )}{6}+\frac {b \,x^{3}}{6 c}+\frac {b \ln \left (c \,x^{3}-1\right )}{12 c^{2}}-\frac {b \ln \left (c \,x^{3}+1\right )}{12 c^{2}}\) | \(57\) |
risch | \(\frac {x^{6} b \ln \left (c \,x^{3}+1\right )}{12}-\frac {x^{6} b \ln \left (-c \,x^{3}+1\right )}{12}+\frac {x^{6} a}{6}+\frac {b \,x^{3}}{6 c}-\frac {b \ln \left (c \,x^{3}+1\right )}{12 c^{2}}+\frac {b \ln \left (c \,x^{3}-1\right )}{12 c^{2}}+\frac {b^{2}}{24 a \,c^{2}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 58, normalized size = 1.35 \begin {gather*} \frac {1}{6} \, a x^{6} + \frac {1}{12} \, {\left (2 \, x^{6} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 54, normalized size = 1.26 \begin {gather*} \frac {2 \, a c^{2} x^{6} + 2 \, b c x^{3} + {\left (b c^{2} x^{6} - b\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{12 \, c^{2}} \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. 181 vs.
\(2 (37) = 74\).
time = 0.42, size = 181, normalized size = 4.21 \begin {gather*} \frac {1}{3} \, c {\left (\frac {{\left (c x^{3} + 1\right )} b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{{\left (c x^{3} - 1\right )} {\left (\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}\right )}} + \frac {\frac {2 \, {\left (c x^{3} + 1\right )} a}{c x^{3} - 1} + \frac {{\left (c x^{3} + 1\right )} b}{c x^{3} - 1} - b}{\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 60, normalized size = 1.40 \begin {gather*} \frac {a\,x^6}{6}+\frac {b\,x^3}{6\,c}+\frac {b\,x^6\,\ln \left (c\,x^3+1\right )}{12}-\frac {b\,x^6\,\ln \left (1-c\,x^3\right )}{12}+\frac {b\,\mathrm {atan}\left (c\,x^3\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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