Optimal. Leaf size=40 \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}-\frac{1}{6} b c \log \left (1-c^2 x^6\right )+b c \log (x) \]
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Rubi [A] time = 0.0270264, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6097, 266, 36, 29, 31} \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}-\frac{1}{6} b c \log \left (1-c^2 x^6\right )+b c \log (x) \]
Antiderivative was successfully verified.
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Rule 6097
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^3\right )}{x^4} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}+(b c) \int \frac{1}{x \left (1-c^2 x^6\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^6\right )\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^6\right )+\frac{1}{6} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^6\right )\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}+b c \log (x)-\frac{1}{6} b c \log \left (1-c^2 x^6\right )\\ \end{align*}
Mathematica [A] time = 0.0122482, size = 45, normalized size = 1.12 \[ -\frac{a}{3 x^3}-\frac{1}{6} b c \log \left (1-c^2 x^6\right )-\frac{b \tanh ^{-1}\left (c x^3\right )}{3 x^3}+b c \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 49, normalized size = 1.2 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b{\it Artanh} \left ( c{x}^{3} \right ) }{3\,{x}^{3}}}+bc\ln \left ( x \right ) -{\frac{bc\ln \left ( c{x}^{3}-1 \right ) }{6}}-{\frac{bc\ln \left ( c{x}^{3}+1 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02135, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{6} \,{\left (c{\left (\log \left (c^{2} x^{6} - 1\right ) - \log \left (x^{6}\right )\right )} + \frac{2 \, \operatorname{artanh}\left (c x^{3}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98002, size = 130, normalized size = 3.25 \begin{align*} -\frac{b c x^{3} \log \left (c^{2} x^{6} - 1\right ) - 6 \, b c x^{3} \log \left (x\right ) + b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13407, size = 69, normalized size = 1.72 \begin{align*} -\frac{1}{6} \, b c \log \left (c^{2} x^{6} - 1\right ) + b c \log \left (x\right ) - \frac{b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{6 \, x^{3}} - \frac{a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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