Optimal. Leaf size=45 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}+\frac{b \tanh ^{-1}\left (\frac{x^2}{c}\right )}{4 c^2}-\frac{b}{4 c x^2} \]
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Rubi [A] time = 0.0320788, antiderivative size = 45, 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, 263, 275, 325, 207} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}+\frac{b \tanh ^{-1}\left (\frac{x^2}{c}\right )}{4 c^2}-\frac{b}{4 c x^2} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 275
Rule 325
Rule 207
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x^5} \, dx &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}-\frac{1}{2} (b c) \int \frac{1}{\left (1-\frac{c^2}{x^4}\right ) x^7} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}-\frac{1}{2} (b c) \int \frac{1}{x^3 \left (-c^2+x^4\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}-\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (-c^2+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{b}{4 c x^2}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-c^2+x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac{b}{4 c x^2}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4}+\frac{b \tanh ^{-1}\left (\frac{x^2}{c}\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.0113542, size = 64, normalized size = 1.42 \[ -\frac{a}{4 x^4}-\frac{b \log \left (x^2-c\right )}{8 c^2}+\frac{b \log \left (c+x^2\right )}{8 c^2}-\frac{b}{4 c x^2}-\frac{b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 57, normalized size = 1.3 \begin{align*} -{\frac{a}{4\,{x}^{4}}}-{\frac{b}{4\,{x}^{4}}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }-{\frac{b}{4\,c{x}^{2}}}-{\frac{b}{8\,{c}^{2}}\ln \left ({\frac{c}{{x}^{2}}}-1 \right ) }+{\frac{b}{8\,{c}^{2}}\ln \left ( 1+{\frac{c}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967116, size = 76, normalized size = 1.69 \begin{align*} \frac{1}{8} \,{\left (c{\left (\frac{\log \left (x^{2} + c\right )}{c^{3}} - \frac{\log \left (x^{2} - c\right )}{c^{3}} - \frac{2}{c^{2} x^{2}}\right )} - \frac{2 \, \operatorname{artanh}\left (\frac{c}{x^{2}}\right )}{x^{4}}\right )} b - \frac{a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54632, size = 109, normalized size = 2.42 \begin{align*} -\frac{2 \, b c x^{2} + 2 \, a c^{2} -{\left (b x^{4} - b c^{2}\right )} \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{8 \, c^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 45.3849, size = 49, normalized size = 1.09 \begin{align*} \begin{cases} - \frac{a}{4 x^{4}} - \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{4 x^{4}} - \frac{b}{4 c x^{2}} + \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\- \frac{a}{4 x^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2143, size = 89, normalized size = 1.98 \begin{align*} \frac{b \log \left (x^{2} + c\right )}{8 \, c^{2}} - \frac{b \log \left (-x^{2} + c\right )}{8 \, c^{2}} - \frac{b \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{8 \, x^{4}} - \frac{b x^{2} + a c}{4 \, c x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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