Optimal. Leaf size=11 \[ \frac {\log (\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2620, 29} \[ \frac {\log (\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2620
Rubi steps
\begin {align*} \int \text {csch}(a+b x) \text {sech}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\log (\tanh (a+b x))}{b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 31, normalized size = 2.82 \[ 2 \left (\frac {\log (\sinh (a+b x))}{2 b}-\frac {\log (\cosh (a+b x))}{2 b}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 60, normalized size = 5.45 \[ -\frac {\log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 41, normalized size = 3.73 \[ -\frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 12, normalized size = 1.09 \[ \frac {\ln \left (\tanh \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 50, normalized size = 4.55 \[ \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 30, normalized size = 2.73 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]
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 \[ \int \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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