Optimal. Leaf size=56 \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]
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Rubi [A] time = 0.0340155, antiderivative size = 56, normalized size of antiderivative = 1., 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 = {6097, 266, 44} \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^2\right )}{x^7} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{3} (b c) \int \frac{1}{x^5 \left (1-c^2 x^4\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^4\right )\\ &=-\frac{b c}{12 x^4}-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{3} b c^3 \log (x)-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0112476, size = 61, normalized size = 1.09 \[ -\frac{a}{6 x^6}-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4}-\frac{b \tanh ^{-1}\left (c x^2\right )}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 63, normalized size = 1.1 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b{\it Artanh} \left ( c{x}^{2} \right ) }{6\,{x}^{6}}}-{\frac{b{c}^{3}\ln \left ( c{x}^{2}+1 \right ) }{12}}-{\frac{bc}{12\,{x}^{4}}}+{\frac{b{c}^{3}\ln \left ( x \right ) }{3}}-{\frac{b{c}^{3}\ln \left ( c{x}^{2}-1 \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975046, size = 69, normalized size = 1.23 \begin{align*} -\frac{1}{12} \,{\left ({\left (c^{2} \log \left (c^{2} x^{4} - 1\right ) - c^{2} \log \left (x^{4}\right ) + \frac{1}{x^{4}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x^{2}\right )}{x^{6}}\right )} b - \frac{a}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11026, size = 150, normalized size = 2.68 \begin{align*} -\frac{b c^{3} x^{6} \log \left (c^{2} x^{4} - 1\right ) - 4 \, b c^{3} x^{6} \log \left (x\right ) + b c x^{2} + b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.4314, size = 97, normalized size = 1.73 \begin{align*} \begin{cases} - \frac{a}{6 x^{6}} + \frac{b c^{3} \log{\left (x \right )}}{3} - \frac{b c^{3} \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{6} - \frac{b c^{3} \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{6} + \frac{b c^{3} \operatorname{atanh}{\left (c x^{2} \right )}}{6} - \frac{b c}{12 x^{4}} - \frac{b \operatorname{atanh}{\left (c x^{2} \right )}}{6 x^{6}} & \text{for}\: c \neq 0 \\- \frac{a}{6 x^{6}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33286, size = 88, normalized size = 1.57 \begin{align*} -\frac{1}{12} \, b c^{3} \log \left (c^{2} x^{4} - 1\right ) + \frac{1}{3} \, b c^{3} \log \left (x\right ) - \frac{b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )}{12 \, x^{6}} - \frac{b c x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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