Optimal. Leaf size=23 \[ \frac {\sinh (a+b x)}{b}-\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]
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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, 203} \[ \frac {\sinh (a+b x)}{b}-\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2592
Rubi steps
\begin {align*} \int \sinh (a+b x) \tanh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac {\sinh (a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b}+\frac {\sinh (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\sinh (a+b x)}{b}-\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 86, normalized size = 3.74 \[ -\frac {4 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - \cosh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2} + 1}{2 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 32, normalized size = 1.39 \[ -\frac {4 \, \arctan \left (e^{\left (b x + a\right )}\right ) - e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 24, normalized size = 1.04 \[ \frac {\sinh \left (b x +a \right )}{b}-\frac {2 \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 41, normalized size = 1.78 \[ \frac {2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 49, normalized size = 2.13 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b} \]
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 \[ \int \sinh {\left (a + b x \right )} \tanh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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