Integrand size = 27, antiderivative size = 57 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\log (\sinh (c+d x))}{a d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a b^2 d}+\frac {\sinh (c+d x)}{b d} \]
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Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2916, 12, 908} \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a b^2 d}+\frac {\log (\sinh (c+d x))}{a d}+\frac {\sinh (c+d x)}{b d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {b \left (-b^2-x^2\right )}{x (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^3 d} \\ & = -\frac {\text {Subst}\left (\int \frac {-b^2-x^2}{x (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^2 d} \\ & = -\frac {\text {Subst}\left (\int \left (-1-\frac {b^2}{a x}+\frac {a^2+b^2}{a (a+x)}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^2 d} \\ & = \frac {\log (\sinh (c+d x))}{a d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a b^2 d}+\frac {\sinh (c+d x)}{b d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {\log (\sinh (c+d x))}{a}-\left (\frac {1}{a}+\frac {a}{b^2}\right ) \log (a+b \sinh (c+d x))+\frac {\sinh (c+d x)}{b}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(57)=114\).
Time = 2.34 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.33
method | result | size |
risch | \(\frac {a x}{b^{2}}+\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 b d}+\frac {2 a c}{b^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d a}\) | \(133\) |
derivativedivides | \(\frac {-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}+\frac {\left (-a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \,b^{2}}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(138\) |
default | \(\frac {-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}+\frac {\left (-a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \,b^{2}}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(138\) |
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.56 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a^{2} d x \cosh \left (d x + c\right ) + a b \cosh \left (d x + c\right )^{2} + a b \sinh \left (d x + c\right )^{2} - a b - 2 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a^{2} d x + a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )\right )}} \]
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\[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\cosh ^{2}{\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (57) = 114\).
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.28 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (d x + c\right )} a}{b^{2} d} + \frac {e^{\left (d x + c\right )}}{2 \, b d} - \frac {e^{\left (-d x - c\right )}}{2 \, b d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a b^{2} d} \]
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Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.65 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}}{b} + \frac {2 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a} - \frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a b^{2}}}{2 \, d} \]
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Time = 1.51 (sec) , antiderivative size = 360, normalized size of antiderivative = 6.32 \[ \int \frac {\cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d}-\frac {\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a\,d}+\frac {\ln \left (4\,a^6+16\,b^6+32\,a^2\,b^4+20\,a^4\,b^2-4\,a^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-16\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-32\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-20\,a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a\,d}+\frac {a\,x}{b^2}-\frac {a\,\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{b^2\,d} \]
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