Integrand size = 26, antiderivative size = 306 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2} \]
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Time = 0.39 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5694, 4269, 3556, 4267, 2317, 2438, 3403, 2296, 2221} \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}+\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}-\frac {(e+f x) \coth (c+d x)}{a d} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rule 3556
Rule 4267
Rule 4269
Rule 5694
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \text {csch}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x) \coth (c+d x)}{a d}-\frac {b \int (e+f x) \text {csch}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f \int \coth (c+d x) \, dx}{a d} \\ & = \frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {\left (2 b^2\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2}+\frac {(b f) \int \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d} \\ & = \frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \sqrt {a^2+b^2}}-\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \sqrt {a^2+b^2}}+\frac {(b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2} \\ & = \frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac {\left (b^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d}+\frac {\left (b^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d} \\ & = \frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \sqrt {a^2+b^2} d^2} \\ & = \frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2} \\ \end{align*}
Time = 5.91 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.14 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a d (e+f x) \coth \left (\frac {1}{2} (c+d x)\right )-2 \left (a f (c+d x)+(a f-b d (e+f x)) \log \left (1-e^{-c-d x}\right )+(a f+b d (e+f x)) \log \left (1+e^{-c-d x}\right )-b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right )-\frac {2 b^2 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+a d (e+f x) \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(283)=566\).
Time = 1.62 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.05
method | result | size |
risch | \(-\frac {2 \left (f x +e \right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {b c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}-\frac {2 b^{2} e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} d \sqrt {a^{2}+b^{2}}}+\frac {2 b^{2} c f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {f b \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}+\frac {f b \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {f \,b^{2} \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {a^{2}+b^{2}}}-\frac {f \,b^{2} \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {a^{2}+b^{2}}}+\frac {f b \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {f \,b^{2} \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {f \,b^{2} \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {f \,b^{2} \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {f \,b^{2} \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {2 f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}\) | \(626\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1830 vs. \(2 (279) = 558\).
Time = 0.31 (sec) , antiderivative size = 1830, normalized size of antiderivative = 5.98 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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