Optimal. Leaf size=41 \[ -\frac {b c}{6 x^3}+\frac {1}{6} b c^2 \tanh ^{-1}\left (c x^3\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, 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, 331,
212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac {1}{6} b c^2 \tanh ^{-1}\left (c x^3\right )-\frac {b c}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rule 331
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{x^7} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac {1}{2} (b c) \int \frac {1}{x^4 \left (1-c^2 x^6\right )} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,x^3\right )\\ &=-\frac {b c}{6 x^3}-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac {1}{6} \left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^3\right )\\ &=-\frac {b c}{6 x^3}+\frac {1}{6} b c^2 \tanh ^{-1}\left (c x^3\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 65, normalized size = 1.59 \begin {gather*} -\frac {a}{6 x^6}-\frac {b c}{6 x^3}-\frac {b \tanh ^{-1}\left (c x^3\right )}{6 x^6}-\frac {1}{12} b c^2 \log \left (1-c x^3\right )+\frac {1}{12} b c^2 \log \left (1+c x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 55, normalized size = 1.34
method | result | size |
default | \(-\frac {a}{6 x^{6}}-\frac {b \arctanh \left (c \,x^{3}\right )}{6 x^{6}}+\frac {b \,c^{2} \ln \left (c \,x^{3}+1\right )}{12}-\frac {b \,c^{2} \ln \left (c \,x^{3}-1\right )}{12}-\frac {b c}{6 x^{3}}\) | \(55\) |
risch | \(-\frac {b \ln \left (c \,x^{3}+1\right )}{12 x^{6}}-\frac {b \,c^{2} \ln \left (c \,x^{3}-1\right ) x^{6}-b \,c^{2} \ln \left (c \,x^{3}+1\right ) x^{6}+2 b c \,x^{3}-b \ln \left (-c \,x^{3}+1\right )+2 a}{12 x^{6}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 51, normalized size = 1.24 \begin {gather*} \frac {1}{12} \, {\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{6}}\right )} b - \frac {a}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 49, normalized size = 1.20 \begin {gather*} -\frac {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 ) + 2 \, a}{12 \, x^{6}} \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 = 0.42, size = 67, normalized size = 1.63 \begin {gather*} \frac {1}{12} \, b c^{2} \log \left (c x^{3} + 1\right ) - \frac {1}{12} \, b c^{2} \log \left (c x^{3} - 1\right ) - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{12 \, x^{6}} - \frac {b c x^{3} + a}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.02, size = 52, normalized size = 1.27 \begin {gather*} \frac {b\,c^2\,\mathrm {atanh}\left (c\,x^3\right )}{6}-\frac {\frac {a}{6}+\frac {b\,\ln \left (c\,x^3+1\right )}{12}-\frac {b\,\ln \left (1-c\,x^3\right )}{12}+\frac {b\,c\,x^3}{6}}{x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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