Optimal. Leaf size=40 \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{2 x^2}-\frac{1}{4} b c \log \left (1-c^2 x^4\right )+b c \log (x) \]
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Rubi [A] time = 0.0255877, 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} \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{2 x^2}-\frac{1}{4} b c \log \left (1-c^2 x^4\right )+b c \log (x) \]
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^2\right )}{x^3} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{2 x^2}+(b c) \int \frac{1}{x \left (1-c^2 x^4\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{2 x^2}+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{2 x^2}+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^4\right )+\frac{1}{4} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^4\right )\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{2 x^2}+b c \log (x)-\frac{1}{4} b c \log \left (1-c^2 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0110482, size = 45, normalized size = 1.12 \[ -\frac{a}{2 x^2}-\frac{1}{4} b c \log \left (1-c^2 x^4\right )-\frac{b \tanh ^{-1}\left (c x^2\right )}{2 x^2}+b c \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 49, normalized size = 1.2 \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b{\it Artanh} \left ( c{x}^{2} \right ) }{2\,{x}^{2}}}-{\frac{bc\ln \left ( c{x}^{2}+1 \right ) }{4}}+bc\ln \left ( x \right ) -{\frac{bc\ln \left ( c{x}^{2}-1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980565, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{4} \,{\left (c{\left (\log \left (c^{2} x^{4} - 1\right ) - \log \left (x^{4}\right )\right )} + \frac{2 \, \operatorname{artanh}\left (c x^{2}\right )}{x^{2}}\right )} b - \frac{a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00197, size = 130, normalized size = 3.25 \begin{align*} -\frac{b c x^{2} \log \left (c^{2} x^{4} - 1\right ) - 4 \, b c x^{2} \log \left (x\right ) + b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.2699, size = 80, normalized size = 2. \begin{align*} \begin{cases} - \frac{a}{2 x^{2}} + b c \log{\left (x \right )} - \frac{b c \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{2} - \frac{b c \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{2} + \frac{b c \operatorname{atanh}{\left (c x^{2} \right )}}{2} - \frac{b \operatorname{atanh}{\left (c x^{2} \right )}}{2 x^{2}} & \text{for}\: c \neq 0 \\- \frac{a}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1719, size = 69, normalized size = 1.72 \begin{align*} -\frac{1}{4} \, b c \log \left (c^{2} x^{4} - 1\right ) + b c \log \left (x\right ) - \frac{b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )}{4 \, x^{2}} - \frac{a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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