Optimal. Leaf size=23 \[ \frac {\cosh (a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2592, 321, 206} \[ \frac {\cosh (a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2592
Rubi steps
\begin {align*} \int \cosh (a+b x) \coth (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\cosh (a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}+\frac {\cosh (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.13 \[ \frac {\cosh (a+b x)}{b}+\frac {\log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 113, normalized size = 4.91 \[ \frac {\cosh \left (b x + a\right )^{2} - 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}{2 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 44, normalized size = 1.91 \[ \frac {e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 21, normalized size = 0.91 \[ \frac {\cosh \left (b x +a \right )-2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 59, normalized size = 2.57 \[ \frac {e^{\left (b x + a\right )}}{2 \, b} + \frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 53, normalized size = 2.30 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b} \]
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 \[ \int \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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