Integrand size = 18, antiderivative size = 108 \[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\frac {(c+d x)^2}{2 (a+b) d}-\frac {b (c+d x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2} \]
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Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3812, 2221, 2317, 2438} \[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=-\frac {b (c+d x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f \left (a^2-b^2\right )}+\frac {b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f^2 \left (a^2-b^2\right )}+\frac {(c+d x)^2}{2 d (a+b)} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3812
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2}{2 (a+b) d}-(2 b) \int \frac {e^{-2 (e+f x)} (c+d x)}{(a+b)^2+\left (-a^2+b^2\right ) e^{-2 (e+f x)}} \, dx \\ & = \frac {(c+d x)^2}{2 (a+b) d}-\frac {b (c+d x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {(b d) \int \log \left (1+\frac {\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right ) f} \\ & = \frac {(c+d x)^2}{2 (a+b) d}-\frac {b (c+d x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}-\frac {(b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (-a^2+b^2\right ) x}{(a+b)^2}\right )}{x} \, dx,x,e^{-2 (e+f x)}\right )}{2 \left (a^2-b^2\right ) f^2} \\ & = \frac {(c+d x)^2}{2 (a+b) d}-\frac {b (c+d x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.41 \[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\frac {1}{2} \left (\frac {2 b (c+d x)^2}{(a+b) d \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right )}-\frac {2 b (c+d x) \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )}{(a-b) (a+b) f}+\frac {b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{(a-b) (a+b) f^2}+\frac {x (2 c+d x) \sinh (e)}{b \cosh (e)+a \sinh (e)}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(107)=214\).
Time = 0.36 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.31
method | result | size |
risch | \(\frac {d \,x^{2}}{2 a +2 b}+\frac {c x}{a +b}+\frac {2 b c \ln \left ({\mathrm e}^{f x +e}\right )}{f \left (a +b \right ) \left (a -b \right )}-\frac {b c \ln \left ({\mathrm e}^{2 f x +2 e} a +b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{f \left (a +b \right ) \left (a -b \right )}+\frac {b d \,x^{2}}{\left (a +b \right ) \left (a -b \right )}-\frac {b d \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right ) x}{f \left (a +b \right ) \left (a -b \right )}+\frac {2 b d e x}{f \left (a +b \right ) \left (a -b \right )}-\frac {b d \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right ) e}{f^{2} \left (a +b \right ) \left (a -b \right )}+\frac {b d \,e^{2}}{f^{2} \left (a +b \right ) \left (a -b \right )}-\frac {b d \operatorname {polylog}\left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right )}{2 f^{2} \left (a +b \right ) \left (a -b \right )}-\frac {2 b d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a +b \right ) \left (a -b \right )}+\frac {b d e \ln \left ({\mathrm e}^{2 f x +2 e} a +b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{f^{2} \left (a +b \right ) \left (a -b \right )}\) | \(357\) |
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Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (103) = 206\).
Time = 0.28 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.78 \[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\frac {{\left (a + b\right )} d f^{2} x^{2} + 2 \, {\left (a + b\right )} c f^{2} x - 2 \, b d {\rm Li}_2\left (\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 2 \, b d {\rm Li}_2\left (-\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) + 2 \, {\left (b d e - b c f\right )} \log \left (2 \, {\left (a + b\right )} \cosh \left (f x + e\right ) + 2 \, {\left (a + b\right )} \sinh \left (f x + e\right ) + 2 \, {\left (a - b\right )} \sqrt {\frac {a + b}{a - b}}\right ) + 2 \, {\left (b d e - b c f\right )} \log \left (2 \, {\left (a + b\right )} \cosh \left (f x + e\right ) + 2 \, {\left (a + b\right )} \sinh \left (f x + e\right ) - 2 \, {\left (a - b\right )} \sqrt {\frac {a + b}{a - b}}\right ) - 2 \, {\left (b d f x + b d e\right )} \log \left (\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 1\right ) - 2 \, {\left (b d f x + b d e\right )} \log \left (-\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 1\right )}{2 \, {\left (a^{2} - b^{2}\right )} f^{2}} \]
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\[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\int \frac {c + d x}{a + b \coth {\left (e + f x \right )}}\, dx \]
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\[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\int { \frac {d x + c}{b \coth \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\int { \frac {d x + c}{b \coth \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {c+d x}{a+b \coth (e+f x)} \, dx=\int \frac {c+d\,x}{a+b\,\mathrm {coth}\left (e+f\,x\right )} \,d x \]
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