Optimal. Leaf size=23 \[ \frac {\text {sech}(a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2622, 321, 207} \[ \frac {\text {sech}(a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2622
Rubi steps
\begin {align*} \int \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=\frac {\text {sech}(a+b x)}{b}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}+\frac {\text {sech}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 1.13 \[ \frac {\text {sech}(a+b x)}{b}+\frac {\log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 155, normalized size = 6.74 \[ -\frac {{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, \cosh \left (b x + a\right ) - 2 \, \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.91, size = 64, normalized size = 2.78 \[ \frac {\frac {4}{e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}} - \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 23, normalized size = 1.00 \[ \frac {\frac {1}{\cosh \left (b x +a \right )}-2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 61, normalized size = 2.65 \[ -\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 52, normalized size = 2.26 \[ \frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]
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 \[ \int \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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