Optimal. Leaf size=46 \[ -\frac {\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}+\frac {\cosh (a+b x)}{b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b} \]
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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 = {5741, 5740,
2718, 3855, 2686, 8} \begin {gather*} -\frac {\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}+\frac {\cosh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2686
Rule 2718
Rule 3855
Rule 5740
Rule 5741
Rubi steps
\begin {align*} \int \coth ^2(c+b x) \sinh (a+b x) \, dx &=\sinh (a-c) \int \coth (c+b x) \text {csch}(c+b x) \, dx+\int \cosh (a+b x) \coth (c+b x) \, dx\\ &=\cosh (a-c) \int \text {csch}(c+b x) \, dx-\frac {(i \sinh (a-c)) \text {Subst}(\int 1 \, dx,x,-i \text {csch}(c+b x))}{b}+\int \sinh (a+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}+\frac {\cosh (a+b x)}{b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 110, normalized size = 2.39 \begin {gather*} -\frac {2 i \text {ArcTan}\left (\frac {(\cosh (c)-\sinh (c)) \left (\cosh (c) \cosh \left (\frac {b x}{2}\right )+\sinh (c) \sinh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \cosh \left (\frac {b x}{2}\right ) \sinh (c)}\right ) \cosh (a-c)}{b}+\frac {\cosh (a) \cosh (b x)}{b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b}+\frac {\sinh (a) \sinh (b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs.
\(2(46)=92\).
time = 1.13, size = 197, normalized size = 4.28
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}{b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs.
\(2 (46) = 92\).
time = 0.27, size = 140, normalized size = 3.04 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1237 vs.
\(2 (46) = 92\).
time = 0.42, size = 1237, normalized size = 26.89 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \sinh {\left (a + b x \right )} \coth ^{2}{\left (b x + c \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (46) = 92\).
time = 0.39, size = 136, normalized size = 2.96 \begin {gather*} -\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 + a + c\right )} + e^{a}\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 + a + c\right )} - e^{a} \right |}\right ) + \frac {2 \, e^{\left (2 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )}}{e^{\left (3 \, b x + 3 \, a + 2 \, c\right )} - e^{\left (b x + 3 \, a\right )}} - e^{\left (b x + a\right )}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.59, size = 181, normalized size = 3.93 \begin {gather*} \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 (2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left (2\,{\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{4\,a-4\,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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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