Optimal. Leaf size=11 \[ \frac{\tan ^{-1}(\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.0616019, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {4380, 203} \[ \frac{\tan ^{-1}(\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 4380
Rule 203
Rubi steps
\begin{align*} \int \frac{\cosh ^2(a+b x)-\sinh ^2(a+b x)}{\cosh ^2(a+b x)+\sinh ^2(a+b x)} \, dx &=\int \frac{1}{\cosh ^2(a+b x)+\sinh ^2(a+b x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{\tan ^{-1}(\tanh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.004128, size = 17, normalized size = 1.55 \[ \frac{\tan ^{-1}(\sinh (2 a+2 b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 148, normalized size = 13.5 \begin{align*} -2\,{\frac{\sqrt{2}}{b \left ( 2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{b \left ( 2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+2\,{\frac{\sqrt{2}}{b \left ( -2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{b \left ( -2+2\,\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\tanh \left ( 1/2\,bx+a/2 \right ) }{-2+2\,\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.75337, size = 66, normalized size = 6. \begin{align*} \frac{\arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-b x - a\right )}\right )}\right ) - \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-b x - a\right )}\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01314, size = 104, normalized size = 9.45 \begin{align*} -\frac{\arctan \left (-\frac{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16033, size = 19, normalized size = 1.73 \begin{align*} \frac{\arctan \left (e^{\left (2 \, b x + 2 \, a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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