Optimal. Leaf size=57 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{b \log \left (c^2-x^4\right )}{12 c^3}+\frac{b \log (x)}{3 c^3}-\frac{b}{12 c x^4} \]
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Rubi [A] time = 0.0400527, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6097, 263, 266, 44} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{b \log \left (c^2-x^4\right )}{12 c^3}+\frac{b \log (x)}{3 c^3}-\frac{b}{12 c x^4} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x^7} \, dx &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{3} (b c) \int \frac{1}{\left (1-\frac{c^2}{x^4}\right ) x^9} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{3} (b c) \int \frac{1}{x^5 \left (-c^2+x^4\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (-c^2+x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4 \left (c^2-x\right )}-\frac{1}{c^2 x^2}-\frac{1}{c^4 x}\right ) \, dx,x,x^4\right )\\ &=-\frac{b}{12 c x^4}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left (c^2-x^4\right )}{12 c^3}\\ \end{align*}
Mathematica [A] time = 0.012913, size = 62, normalized size = 1.09 \[ -\frac{a}{6 x^6}-\frac{b \log \left (x^4-c^2\right )}{12 c^3}+\frac{b \log (x)}{3 c^3}-\frac{b}{12 c x^4}-\frac{b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 45, normalized size = 0.8 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b}{6\,{x}^{6}}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }-{\frac{b}{12\,c{x}^{4}}}-{\frac{b}{12\,{c}^{3}}\ln \left ({\frac{{c}^{2}}{{x}^{4}}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979566, size = 74, normalized size = 1.3 \begin{align*} -\frac{1}{12} \,{\left (c{\left (\frac{\log \left (x^{4} - c^{2}\right )}{c^{4}} - \frac{\log \left (x^{4}\right )}{c^{4}} + \frac{1}{c^{2} x^{4}}\right )} + \frac{2 \, \operatorname{artanh}\left (\frac{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 = 1.67821, size = 151, normalized size = 2.65 \begin{align*} -\frac{b x^{6} \log \left (x^{4} - c^{2}\right ) - 4 \, b x^{6} \log \left (x\right ) + b c^{2} x^{2} + b c^{3} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{3}}{12 \, c^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 47.1008, size = 94, normalized size = 1.65 \begin{align*} \begin{cases} - \frac{a}{6 x^{6}} - \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{6 x^{6}} - \frac{b}{12 c x^{4}} + \frac{b \log{\left (x \right )}}{3 c^{3}} - \frac{b \log{\left (- i \sqrt{c} + x \right )}}{6 c^{3}} - \frac{b \log{\left (i \sqrt{c} + x \right )}}{6 c^{3}} + \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{6 c^{3}} & \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 = 1.16046, size = 88, normalized size = 1.54 \begin{align*} -\frac{b \log \left (x^{4} - c^{2}\right )}{12 \, c^{3}} + \frac{b \log \left (x\right )}{3 \, c^{3}} - \frac{b \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{12 \, x^{6}} - \frac{b x^{2} + 2 \, a c}{12 \, c x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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