Integrand size = 30, antiderivative size = 286 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2} \]
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Time = 0.41 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {5704, 5558, 3377, 2717, 4267, 2317, 2438, 5684, 3403, 2296, 2221} \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {f \sqrt {a^2+b^2} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {f \sqrt {a^2+b^2} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a b d}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {e x}{b}+\frac {f x^2}{2 b} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2717
Rule 3377
Rule 3403
Rule 4267
Rule 5558
Rule 5684
Rule 5704
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \cosh (c+d x) \coth (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = \frac {\int (e+f x) \text {csch}(c+d x) \, dx}{a}+\frac {\int (e+f x) \, dx}{b}-\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{a b} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\left (2 \left (a^2+b^2\right )\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 b}-\frac {f \int \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {f \int \log \left (1+e^{c+d x}\right ) \, dx}{a d} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\left (2 \sqrt {a^2+b^2}\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}+\frac {\left (2 \sqrt {a^2+b^2}\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}-\frac {f \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}+\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {\left (\sqrt {a^2+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 b d}-\frac {\left (\sqrt {a^2+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 b d} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {\left (\sqrt {a^2+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 b d^2}-\frac {\left (\sqrt {a^2+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 b d^2} \\ & = \frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.04 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-a (c+d x) (c f-d (2 e+f x))+2 b \left (d (e+f x) \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )+f \operatorname {PolyLog}\left (2,e^{c+d x}\right )\right )+2 \sqrt {a^2+b^2} \left (2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a b d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(969\) vs. \(2(261)=522\).
Time = 1.95 (sec) , antiderivative size = 970, normalized size of antiderivative = 3.39
method | result | size |
risch | \(\frac {f \,x^{2}}{2 b}+\frac {e x}{b}+\frac {2 a e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b d \sqrt {a^{2}+b^{2}}}+\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}-\frac {2 b c f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {2 c a f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \,d^{2} \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d a}-\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}+\frac {2 b e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d a \sqrt {a^{2}+b^{2}}}-\frac {c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{b \,d^{2} \sqrt {a^{2}+b^{2}}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{b \,d^{2} \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}-\frac {b f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{b d \sqrt {a^{2}+b^{2}}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{b d \sqrt {a^{2}+b^{2}}}-\frac {a f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{b \,d^{2} \sqrt {a^{2}+b^{2}}}+\frac {a f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{b \,d^{2} \sqrt {a^{2}+b^{2}}}\) | \(970\) |
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Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (257) = 514\).
Time = 0.31 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.09 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a d^{2} f x^{2} + 2 \, a d^{2} e x - 2 \, b f \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, b f \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, b f {\rm Li}_2\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - 2 \, b f {\rm Li}_2\left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )\right ) + 2 \, {\left (b d e - b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (b d e - b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (b d f x + b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, {\left (b d f x + b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left (b d f x + b d e\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (b d e - b c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (b d f x + b c f\right )} \log \left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) + 1\right )}{2 \, a b d^{2}} \]
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\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh {\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \coth \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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