Optimal. Leaf size=46 \[ -\frac {\cosh (a-c) \text {csch}(b x+c)}{b}-\frac {\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}+\frac {\sinh (a+b x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 46, 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 = {5621, 5622, 2637, 3770, 2606, 8} \[ -\frac {\cosh (a-c) \text {csch}(b x+c)}{b}-\frac {\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}+\frac {\sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 2637
Rule 3770
Rule 5621
Rule 5622
Rubi steps
\begin {align*} \int \cosh (a+b x) \coth ^2(c+b x) \, dx &=\cosh (a-c) \int \coth (c+b x) \text {csch}(c+b x) \, dx+\int \coth (c+b x) \sinh (a+b x) \, dx\\ &=-\frac {(i \cosh (a-c)) \operatorname {Subst}(\int 1 \, dx,x,-i \text {csch}(c+b x))}{b}+\sinh (a-c) \int \text {csch}(c+b x) \, dx+\int \cosh (a+b x) \, dx\\ &=-\frac {\cosh (a-c) \text {csch}(c+b x)}{b}-\frac {\tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{b}+\frac {\sinh (a+b x)}{b}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 110, normalized size = 2.39 \[ -\frac {\cosh (a-c) \text {csch}(b x+c)}{b}-\frac {2 i \sinh (a-c) \tan ^{-1}\left (\frac {(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac {b x}{2}\right )+\cosh (c) \cosh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \sinh (c) \cosh \left (\frac {b x}{2}\right )}\right )}{b}+\frac {\sinh (a) \cosh (b x)}{b}+\frac {\cosh (a) \sinh (b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 1237, normalized size = 26.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 125, normalized size = 2.72 \[ -\frac {{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) - {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) + \frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, c\right )} - 1\right )} e^{\left (-a\right )}}{e^{\left (3 \, b x + 2 \, c\right )} - e^{\left (b x\right )}} - e^{\left (b x + a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 195, normalized size = 4.24 \[ \frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}+{\mathrm e}^{2 c}\right )}{b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 144, normalized size = 3.13 \[ -\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (3 \, e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (2 \, c\right )}}{2 \, b {\left (e^{\left (-b x - a + 2 \, c\right )} - e^{\left (-3 \, b x - a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 183, normalized size = 3.98 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {\mathrm {atan}\left (-\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}-2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left ({\mathrm {e}}^{4\,a-4\,c}-2\,{\mathrm {e}}^{2\,a-2\,c}+1\right )}}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}+1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}-{\mathrm {e}}^{2\,a+2\,b\,x}\right )} \]
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 \[ \int \cosh {\left (a + b x \right )} \coth ^{2}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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