Optimal. Leaf size=51 \[ \frac{\tan ^{-1}\left (1-\sqrt{2} \tanh (a+b x)\right )}{\sqrt{2} b}-\frac{\tan ^{-1}\left (\sqrt{2} \tanh (a+b x)+1\right )}{\sqrt{2} b} \]
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Rubi [A] time = 1.41791, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1162, 617, 204} \[ \frac{\tan ^{-1}\left (1-\sqrt{2} \tanh (a+b x)\right )}{\sqrt{2} b}-\frac{\tan ^{-1}\left (\sqrt{2} \tanh (a+b x)+1\right )}{\sqrt{2} b} \]
Antiderivative was successfully verified.
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Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{-\text{csch}^4(a+b x)+\text{sech}^4(a+b x)}{\text{csch}^4(a+b x)+\text{sech}^4(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1-x^2}{1+x^4} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\tanh (a+b x)\right )}{2 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\tanh (a+b x)\right )}{2 b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \tanh (a+b x)\right )}{\sqrt{2} b}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \tanh (a+b x)\right )}{\sqrt{2} b}\\ &=\frac{\tan ^{-1}\left (1-\sqrt{2} \tanh (a+b x)\right )}{\sqrt{2} b}-\frac{\tan ^{-1}\left (1+\sqrt{2} \tanh (a+b x)\right )}{\sqrt{2} b}\\ \end{align*}
Mathematica [A] time = 0.0258839, size = 26, normalized size = 0.51 \[ -\frac{\tan ^{-1}\left (\frac{\sinh (2 a+2 b x)}{\sqrt{2}}\right )}{\sqrt{2} b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.134, size = 138, normalized size = 2.7 \begin{align*}{\frac{{\frac{i}{4}}\sqrt{2}}{b}\ln \left ( 2\,i\sqrt{2} \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{3}+ \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{4}+2\,i\sqrt{2}\tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) -2\, \left ( \tanh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1 \right ) }-{\frac{{\frac{i}{4}}\sqrt{2}}{b}\ln \left ( -2\,i\sqrt{2} \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{3}+ \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{4}-2\,i\sqrt{2}\tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) -2\, \left ( \tanh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+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*} -2 \, \int \frac{{\left (e^{\left (-b x - a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\right )} e^{\left (-b x - a\right )}}{6 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1}\,{d x} - 2 \, \int \frac{{\left (e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{6 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70971, size = 558, normalized size = 10.94 \begin{align*} \frac{\sqrt{2} \arctan \left (-\frac{\sqrt{2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt{2} \sinh \left (b x + a\right )^{3} +{\left (3 \, \sqrt{2} \cosh \left (b x + a\right )^{2} - 7 \, \sqrt{2}\right )} \sinh \left (b x + a\right ) + 7 \, \sqrt{2} \cosh \left (b x + a\right )}{4 \,{\left (\cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - \sinh \left (b x + a\right )^{3}\right )}}\right ) + \sqrt{2} \arctan \left (-\frac{\sqrt{2} \cosh \left (b x + a\right ) + \sqrt{2} \sinh \left (b x + a\right )}{4 \,{\left (\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}}\right )}{2 \, b} \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}^{4}{\left (a + b x \right )}}{\operatorname{csch}^{4}{\left (a + b x \right )} + \operatorname{sech}^{4}{\left (a + b x \right )}}\, dx - \int - \frac{\operatorname{sech}^{4}{\left (a + b x \right )}}{\operatorname{csch}^{4}{\left (a + b x \right )} + \operatorname{sech}^{4}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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