Optimal. Leaf size=47 \[ \frac{1}{3 b (\tanh (a+b x)+1)}+\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (a+b x)}{\sqrt{3}}\right )}{3 \sqrt{3} b} \]
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Rubi [A] time = 0.40999, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2074, 618, 204} \[ \frac{1}{3 b (\tanh (a+b x)+1)}+\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (a+b x)}{\sqrt{3}}\right )}{3 \sqrt{3} b} \]
Antiderivative was successfully verified.
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Rule 2074
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{-\text{csch}^3(a+b x)+\text{sech}^3(a+b x)}{\text{csch}^3(a+b x)+\text{sech}^3(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1-x-x^2}{1+x+x^3+x^4} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{3 (1+x)^2}-\frac{2}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{1}{3 b (1+\tanh (a+b x))}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tanh (a+b x)\right )}{3 b}\\ &=\frac{1}{3 b (1+\tanh (a+b x))}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tanh (a+b x)\right )}{3 b}\\ &=\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (a+b x)}{\sqrt{3}}\right )}{3 \sqrt{3} b}+\frac{1}{3 b (1+\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.318515, size = 52, normalized size = 1.11 \[ \frac{-3 \sinh (2 (a+b x))+3 \cosh (2 (a+b x))-8 \sqrt{3} \tan ^{-1}\left (\frac{2 \tanh (a+b x)-1}{\sqrt{3}}\right )}{18 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.247, size = 120, normalized size = 2.6 \begin{align*}{\frac{2}{3\,b} \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) ^{-2}}-{\frac{2}{3\,b} \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) ^{-1}}+{\frac{{\frac{2\,i}{9}}\sqrt{3}}{b}\ln \left ( \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}+ \left ( i\sqrt{3}-1 \right ) \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) }-{\frac{{\frac{2\,i}{9}}\sqrt{3}}{b}\ln \left ( \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}+ \left ( -i\sqrt{3}-1 \right ) \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.77548, size = 126, normalized size = 2.68 \begin{align*} -\frac{4 \,{\left (\sqrt{3} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, \sqrt{3} e^{\left (-b x - a\right )} + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \sqrt{3} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, \sqrt{3} e^{\left (-b x - a\right )} - 3^{\frac{1}{4}} \sqrt{2}\right )}\right )\right )}}{9 \, b} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04038, size = 369, normalized size = 7.85 \begin{align*} \frac{8 \,{\left (\sqrt{3} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt{3} \sinh \left (b x + a\right )^{2}\right )} \arctan \left (-\frac{\sqrt{3} \cosh \left (b x + a\right ) + \sqrt{3} \sinh \left (b x + a\right )}{3 \,{\left (\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}}\right ) + 3}{18 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\operatorname{csch}^{3}{\left (a + b x \right )}}{\operatorname{csch}^{3}{\left (a + b x \right )} + \operatorname{sech}^{3}{\left (a + b x \right )}}\, dx - \int - \frac{\operatorname{sech}^{3}{\left (a + b x \right )}}{\operatorname{csch}^{3}{\left (a + b x \right )} + \operatorname{sech}^{3}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1638, size = 51, normalized size = 1.09 \begin{align*} -\frac{4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} e^{\left (2 \, b x + 2 \, a\right )}\right )}{9 \, b} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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