Integrand size = 19, antiderivative size = 64 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\text {arctanh}(\cosh (c+d x))}{a d}+\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \]
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Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2826, 3855, 2739, 632, 210} \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}-\frac {\text {arctanh}(\cosh (c+d x))}{a d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2826
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {\text {arctanh}(\cosh (c+d x))}{a d}+\frac {(2 i b) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a d} \\ & = -\frac {\text {arctanh}(\cosh (c+d x))}{a d}-\frac {(4 i b) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a d} \\ & = -\frac {\text {arctanh}(\cosh (c+d x))}{a d}+\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.28 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
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Time = 1.00 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {2 b \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(63\) |
default | \(\frac {-\frac {2 b \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(63\) |
risch | \(\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, d a}-\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (61) = 122\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.48 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} b \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + {\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{{\left (a^{3} + a b^{2}\right )} d} \]
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\[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a d} - \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \]
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Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.59 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {b \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a} + \frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a}}{d} \]
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Time = 1.20 (sec) , antiderivative size = 347, normalized size of antiderivative = 5.42 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\ln \left (32\,a-32\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a\,d}-\frac {\ln \left (32\,a+32\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a\,d}+\frac {b\,\ln \left (128\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a^2\,b^3-64\,a^4\,b+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,a^3\,b\,\sqrt {a^2+b^2}+160\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-96\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{d\,a^3+d\,a\,b^2}-\frac {b\,\ln \left (64\,a^4\,b+64\,a^2\,b^3-128\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,a^3\,b\,\sqrt {a^2+b^2}-160\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-96\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{d\,a^3+d\,a\,b^2} \]
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