Optimal. Leaf size=65 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}-\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{2 b}{3 c x} \]
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Rubi [A] time = 0.0371315, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6097, 263, 325, 298, 203, 206} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}-\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{2 b}{3 c x} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 325
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x^4} \, dx &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}-\frac{1}{3} (2 b c) \int \frac{1}{\left (1-\frac{c^2}{x^4}\right ) x^6} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}-\frac{1}{3} (2 b c) \int \frac{1}{x^2 \left (-c^2+x^4\right )} \, dx\\ &=-\frac{2 b}{3 c x}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}-\frac{(2 b) \int \frac{x^2}{-c^2+x^4} \, dx}{3 c}\\ &=-\frac{2 b}{3 c x}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}+\frac{b \int \frac{1}{c-x^2} \, dx}{3 c}-\frac{b \int \frac{1}{c+x^2} \, dx}{3 c}\\ &=-\frac{2 b}{3 c x}-\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0305724, size = 90, normalized size = 1.38 \[ -\frac{a}{3 x^3}-\frac{b \log \left (\sqrt{c}-x\right )}{6 c^{3/2}}+\frac{b \log \left (\sqrt{c}+x\right )}{6 c^{3/2}}-\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b}{3 c x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 55, normalized size = 0.9 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b}{3\,{x}^{3}}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }-{\frac{2\,b}{3\,cx}}-{\frac{b}{3}\arctan \left ({x{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{b}{3}{\it Artanh} \left ({\frac{1}{x}\sqrt{c}} \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84796, size = 451, normalized size = 6.94 \begin{align*} \left [-\frac{2 \, b \sqrt{c} x^{3} \arctan \left (\frac{x}{\sqrt{c}}\right ) - b \sqrt{c} x^{3} \log \left (\frac{x^{2} + 2 \, \sqrt{c} x + c}{x^{2} - c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}, -\frac{2 \, b \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-c} x}{c}\right ) + b \sqrt{-c} x^{3} \log \left (\frac{x^{2} + 2 \, \sqrt{-c} x - c}{x^{2} + c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.6988, size = 706, normalized size = 10.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30144, size = 97, normalized size = 1.49 \begin{align*} -\frac{1}{3} \, b{\left (\frac{\arctan \left (\frac{x}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{\arctan \left (\frac{x}{\sqrt{c}}\right )}{c^{\frac{3}{2}}}\right )} - \frac{b \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{6 \, x^{3}} - \frac{2 \, b x^{2} + a c}{3 \, c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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