Optimal. Leaf size=45 \[ -\frac {\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}+\frac {\cosh (a-c) \text {sech}(b x+c)}{b}+\frac {\cosh (a+b x)}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5620, 5623, 2638, 3770, 2606, 8} \[ -\frac {\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}+\frac {\cosh (a-c) \text {sech}(b x+c)}{b}+\frac {\cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 2638
Rule 3770
Rule 5620
Rule 5623
Rubi steps
\begin {align*} \int \sinh (a+b x) \tanh ^2(c+b x) \, dx &=-(\cosh (a-c) \int \text {sech}(c+b x) \tanh (c+b x) \, dx)+\int \cosh (a+b x) \tanh (c+b x) \, dx\\ &=\frac {\cosh (a-c) \operatorname {Subst}(\int 1 \, dx,x,\text {sech}(c+b x))}{b}-\sinh (a-c) \int \text {sech}(c+b x) \, dx+\int \sinh (a+b x) \, dx\\ &=\frac {\cosh (a+b x)}{b}+\frac {\cosh (a-c) \text {sech}(c+b x)}{b}-\frac {\tan ^{-1}(\sinh (c+b x)) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 102, normalized size = 2.27 \[ \frac {\cosh (a-c) \text {sech}(b x+c)}{b}-\frac {2 \sinh (a-c) \tan ^{-1}\left (\frac {(\cosh (c)-\sinh (c)) \left (\sinh (c) \cosh \left (\frac {b x}{2}\right )+\cosh (c) \sinh \left (\frac {b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac {b x}{2}\right )-\sinh (c) \cosh \left (\frac {b x}{2}\right )}\right )}{b}+\frac {\sinh (a) \sinh (b x)}{b}+\frac {\cosh (a) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 902, normalized size = 20.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 88, normalized size = 1.96 \[ -\frac {2 \, {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - \frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, c\right )} + 1\right )} e^{\left (-a\right )}}{e^{\left (3 \, b x + 2 \, c\right )} + e^{\left (b x\right )}} - e^{\left (b x + a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 205, normalized size = 4.56 \[ \frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}+{\mathrm e}^{2 c}\right )}{b \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 105, normalized size = 2.33 \[ \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} + \frac {e^{\left (-b x - a\right )}}{2 \, b} + \frac {{\left (3 \, e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (2 \, c\right )}}{2 \, b {\left (e^{\left (-b x - a + 2 \, c\right )} + e^{\left (-3 \, b x - a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 173, normalized size = 3.84 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {b^2}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}-2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left ({\mathrm {e}}^{4\,a-4\,c}-2\,{\mathrm {e}}^{2\,a-2\,c}+1\right )}}{\sqrt {b^2}}+\frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}+1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{2\,a+2\,b\,x}\right )} \]
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 \[ \int \sinh {\left (a + b x \right )} \tanh ^{2}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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