Optimal. Leaf size=12 \[ -\frac {\tan ^{-1}(\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.26, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {204} \[ -\frac {\tan ^{-1}(\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 204
Rubi steps
\begin {align*} \int \frac {-\text {csch}^2(a+b x)+\text {sech}^2(a+b x)}{\text {csch}^2(a+b x)+\text {sech}^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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.42 \[ -\frac {\tan ^{-1}(\sinh (2 a+2 b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 37, normalized size = 3.08 \[ \frac {\arctan \left (-\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 43, normalized size = 3.58 \[ \frac {\arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (b x + a\right )}\right )}\right ) - \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (b x + a\right )}\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.80, size = 148, normalized size = 12.33 \[ -\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )}{-2+2 \sqrt {2}}\right )}{b \left (-2+2 \sqrt {2}\right )}+\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )}{-2+2 \sqrt {2}}\right )}{b \left (-2+2 \sqrt {2}\right )}+\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )}{2+2 \sqrt {2}}\right )}{b \left (2+2 \sqrt {2}\right )}+\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )}{2+2 \sqrt {2}}\right )}{b \left (2+2 \sqrt {2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 50, normalized size = 4.17 \[ -\frac {\arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-b x - a\right )}\right )}\right ) - \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-b x - a\right )}\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.96, size = 26, normalized size = 2.17 \[ -\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {csch}^{2}{\left (a + b x \right )}}{\operatorname {csch}^{2}{\left (a + b x \right )} + \operatorname {sech}^{2}{\left (a + b x \right )}}\, dx - \int \left (- \frac {\operatorname {sech}^{2}{\left (a + b x \right )}}{\operatorname {csch}^{2}{\left (a + b x \right )} + \operatorname {sech}^{2}{\left (a + b x \right )}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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