Integrand size = 30, antiderivative size = 243 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2} \]
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Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5706, 5560, 3855, 5688, 3797, 2221, 2317, 2438, 5680} \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d}+\frac {b (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 3855
Rule 5560
Rule 5680
Rule 5688
Rule 5706
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x) \text {csch}(c+d x)}{a d}-\frac {b \int (e+f x) \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f \int \text {csch}(c+d x) \, dx}{a d} \\ & = -\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {(2 b) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2} \\ & = -\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {(b f) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {(b f) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d} \\ & = -\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {(b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2} \\ & = -\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(621\) vs. \(2(243)=486\).
Time = 8.15 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.56 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (-d e \cosh \left (\frac {1}{2} (c+d x)\right )+c f \cosh \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{2 a d^2}+\frac {-\frac {b (d e-c f+f (c+d x))^2}{2 f}+(-b d e+a f+b c f-b f (c+d x)) \log \left (1-e^{-c-d x}\right )+(-b d e-a f+b c f-b f (c+d x)) \log \left (1+e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a^2 d^2}+\frac {b \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a^2 d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (d e \sinh \left (\frac {1}{2} (c+d x)\right )-c f \sinh \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(229)=458\).
Time = 2.33 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.17
method | result | size |
risch | \(-\frac {2 \left (f x +e \right ) {\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {b e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d}+\frac {f b \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}+\frac {f b \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}+\frac {f b \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}+\frac {f b \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}-\frac {f b \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}-\frac {b c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {f b \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {f b \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}+\frac {f b \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}+\frac {f b \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}\) | \(528\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1221 vs. \(2 (226) = 452\).
Time = 0.29 (sec) , antiderivative size = 1221, normalized size of antiderivative = 5.02 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \coth {\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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