Optimal. Leaf size=27 \[ \frac {\sinh ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2590, 14} \[ \frac {\sinh ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2590
Rubi steps
\begin {align*} \int \cosh ^2(a+b x) \coth (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac {\log (\sinh (a+b x))}{b}+\frac {\sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.93 \[ \frac {\sinh ^2(a+b x)+2 \log (\sinh (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 203, normalized size = 7.52 \[ -\frac {8 \, b x \cosh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{4} - 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - \sinh \left (b x + a\right )^{4} + 2 \, {\left (4 \, b x - 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - 8 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (4 \, b x \cosh \left (b x + a\right ) - \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right ) - 1}{8 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 63, normalized size = 2.33 \[ -\frac {8 \, b x - {\left (4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 8 \, a - e^{\left (2 \, b x + 2 \, a\right )} - 8 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 26, normalized size = 0.96 \[ \frac {\cosh ^{2}\left (b x +a \right )}{2 b}+\frac {\ln \left (\sinh \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 70, normalized size = 2.59 \[ \frac {b x + a}{b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 49, normalized size = 1.81 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x+\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{8\,b} \]
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 \[ \int \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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