Optimal. Leaf size=95 \[ -\frac{2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac{2 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac{\sinh (x)}{3 (1-\cosh (x))} \]
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Rubi [A] time = 0.121, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3213, 2648, 2659, 208, 205} \[ -\frac{2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac{2 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac{\sinh (x)}{3 (1-\cosh (x))} \]
Antiderivative was successfully verified.
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Rule 3213
Rule 2648
Rule 2659
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{1-\cosh ^3(x)} \, dx &=\int \left (\frac{1}{3 (1-\cosh (x))}+\frac{1}{3 \left (1+\sqrt [3]{-1} \cosh (x)\right )}+\frac{1}{3 \left (1-(-1)^{2/3} \cosh (x)\right )}\right ) \, dx\\ &=\frac{1}{3} \int \frac{1}{1-\cosh (x)} \, dx+\frac{1}{3} \int \frac{1}{1+\sqrt [3]{-1} \cosh (x)} \, dx+\frac{1}{3} \int \frac{1}{1-(-1)^{2/3} \cosh (x)} \, dx\\ &=-\frac{\sinh (x)}{3 (1-\cosh (x))}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{-1}-\left (1-\sqrt [3]{-1}\right ) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-(-1)^{2/3}-\left (1+(-1)^{2/3}\right ) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=-\frac{2 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac{2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac{\sinh (x)}{3 (1-\cosh (x))}\\ \end{align*}
Mathematica [C] time = 0.547871, size = 147, normalized size = 1.55 \[ \frac{1}{3} \coth \left (\frac{x}{2}\right )+\frac{\left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{\left (1-i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{2 \left (3-i \sqrt{3}\right )}}\right )}{3 \sqrt{\frac{3}{2} \left (3-i \sqrt{3}\right )}}+\frac{\left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{\left (1+i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{2 \left (3+i \sqrt{3}\right )}}\right )}{3 \sqrt{\frac{3}{2} \left (3+i \sqrt{3}\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 212, normalized size = 2.2 \begin{align*}{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{\sqrt [4]{3}\sqrt{2}}{6}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+1 \right ) }+{\frac{\sqrt [4]{3}\sqrt{2}}{6}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }-1 \right ) }+{\frac{\sqrt [4]{3}\sqrt{2}}{12}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) ^{-1}} \right ) }-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{36}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) ^{-1}} \right ) }-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{18}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+1 \right ) }-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{18}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{3 \,{\left (e^{x} - 1\right )}} + \int \frac{2 \,{\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + e^{x}\right )}}{3 \,{\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.50265, size = 1924, normalized size = 20.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.28013, size = 405, normalized size = 4.26 \begin{align*} \frac{\sqrt{2} \sqrt [4]{3} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} - 4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} + 4 \sqrt{3} \right )} \tanh{\left (\frac{x}{2} \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{\sqrt{2} \sqrt [4]{3} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} + 4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} + 4 \sqrt{3} \right )} \tanh{\left (\frac{x}{2} \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} - 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} - 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} + 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} + 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{6}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} + \frac{2 \sqrt{3}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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