Integrand size = 18, antiderivative size = 261 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=3 a b^2 c x+\frac {b^3 d x}{2 f}+\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}-\frac {b^3 d \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {b^3 d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2} \]
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Time = 0.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3803, 3797, 2221, 2317, 2438, 3801, 3556, 3554, 8} \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\frac {a^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {3 a^2 b (c+d x)^2}{2 d}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}+3 a b^2 c x+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac {3}{2} a b^2 d x^2+\frac {b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}-\frac {b^3 (c+d x)^2}{2 d}+\frac {b^3 d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {b^3 d \coth (e+f x)}{2 f^2}+\frac {b^3 d x}{2 f} \]
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3554
Rule 3556
Rule 3797
Rule 3801
Rule 3803
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 (c+d x)+3 a^2 b (c+d x) \coth (e+f x)+3 a b^2 (c+d x) \coth ^2(e+f x)+b^3 (c+d x) \coth ^3(e+f x)\right ) \, dx \\ & = \frac {a^3 (c+d x)^2}{2 d}+\left (3 a^2 b\right ) \int (c+d x) \coth (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x) \coth ^2(e+f x) \, dx+b^3 \int (c+d x) \coth ^3(e+f x) \, dx \\ & = \frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}-\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x) \, dx+b^3 \int (c+d x) \coth (e+f x) \, dx+\frac {\left (3 a b^2 d\right ) \int \coth (e+f x) \, dx}{f}+\frac {\left (b^3 d\right ) \int \coth ^2(e+f x) \, dx}{2 f} \\ & = 3 a b^2 c x+\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}-\frac {b^3 d \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}-\left (2 b^3\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx-\frac {\left (3 a^2 b d\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}+\frac {\left (b^3 d\right ) \int 1 \, dx}{2 f} \\ & = 3 a b^2 c x+\frac {b^3 d x}{2 f}+\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}-\frac {b^3 d \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (b^3 d\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f} \\ & = 3 a b^2 c x+\frac {b^3 d x}{2 f}+\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}-\frac {b^3 d \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (b^3 d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2} \\ & = 3 a b^2 c x+\frac {b^3 d x}{2 f}+\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}-\frac {b^3 d \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac {b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {b^3 d \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2} \\ \end{align*}
Time = 8.18 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.22 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\frac {(a+b \coth (e+f x))^3 \sinh (e+f x) \left (-b^3 f (c+d x)-a \left (a^2+3 b^2\right ) (e+f x) (-2 c f+d (e-f x)) \sinh ^2(e+f x)+2 b \left (\frac {3 a^2 f^2 (c+d x)^2}{2 d}+\frac {b^2 f^2 (c+d x)^2}{2 d}+3 a b d (e+f x)+\left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \log \left (1-e^{-e-f x}\right )+\left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \log \left (1+e^{-e-f x}\right )-\left (3 a^2+b^2\right ) d \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )-\left (3 a^2+b^2\right ) d \operatorname {PolyLog}\left (2,e^{-e-f x}\right )\right ) \sinh ^2(e+f x)-\frac {1}{2} b^2 (b d+6 a f (c+d x)) \sinh (2 (e+f x))\right )}{2 f^2 (b \cosh (e+f x)+a \sinh (e+f x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(650\) vs. \(2(243)=486\).
Time = 0.49 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.49
method | result | size |
risch | \(-\frac {b^{3} e d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {3 b^{2} d a \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {3 b^{2} d a \ln \left (1+{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b d \,a^{2} \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b d \,a^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {b^{3} d \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {b^{3} d \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{f}-\frac {3 a^{2} b d \,x^{2}}{2}+3 a^{2} b c x +\frac {b^{3} d \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 b \,a^{2} d e x}{f}+\frac {6 b e \,a^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {b^{3} d \,e^{2}}{f^{2}}-\frac {2 b^{3} c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b^{3} c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {b^{3} c \ln \left (1+{\mathrm e}^{f x +e}\right )}{f}+\frac {b^{3} d \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+3 a \,b^{2} c x +\frac {3 a \,b^{2} d \,x^{2}}{2}-\frac {2 b^{3} d e x}{f}-\frac {3 b \,a^{2} d \,e^{2}}{f^{2}}-\frac {6 b \,a^{2} c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {2 b^{3} e d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 b^{2} d a \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {b^{2} \left (6 a d f x \,{\mathrm e}^{2 f x +2 e}+2 b d f x \,{\mathrm e}^{2 f x +2 e}+6 a c f \,{\mathrm e}^{2 f x +2 e}+2 b c f \,{\mathrm e}^{2 f x +2 e}-6 a d f x +{\mathrm e}^{2 f x +2 e} d b -6 a c f -b d \right )}{f^{2} \left ({\mathrm e}^{2 f x +2 e}-1\right )^{2}}+\frac {b^{3} d \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {3 b \,a^{2} c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {3 b \,a^{2} c \ln \left (1+{\mathrm e}^{f x +e}\right )}{f}+\frac {a^{3} d \,x^{2}}{2}-\frac {b^{3} d \,x^{2}}{2}+a^{3} c x +b^{3} c x -\frac {3 b d \,a^{2} e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {3 b d \,a^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {3 b d \,a^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {3 b d \,a^{2} \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{f}\) | \(651\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2907 vs. \(2 (241) = 482\).
Time = 0.30 (sec) , antiderivative size = 2907, normalized size of antiderivative = 11.14 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \]
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\[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\int \left (a + b \coth {\left (e + f x \right )}\right )^{3} \left (c + d x\right )\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (241) = 482\).
Time = 0.27 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.02 \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\frac {1}{2} \, a^{3} d x^{2} + a^{3} c x + \frac {3 \, a^{2} b c \log \left (\sinh \left (f x + e\right )\right )}{f} - {\left (3 \, a^{2} b d + b^{3} d\right )} x^{2} - \frac {2 \, {\left (b^{3} c f + 3 \, a b^{2} d\right )} x}{f} + \frac {12 \, a b^{2} c f + 2 \, b^{3} d + {\left (3 \, a^{2} b d f^{2} + 3 \, a b^{2} d f^{2} + b^{3} d f^{2}\right )} x^{2} + 2 \, {\left (b^{3} c f^{2} + 3 \, {\left (c f^{2} + 2 \, d f\right )} a b^{2}\right )} x + {\left ({\left (3 \, a^{2} b d f^{2} e^{\left (4 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (4 \, e\right )} + b^{3} d f^{2} e^{\left (4 \, e\right )}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c f^{2} e^{\left (4 \, e\right )} + b^{3} c f^{2} e^{\left (4 \, e\right )}\right )} x\right )} e^{\left (4 \, f x\right )} - 2 \, {\left (6 \, a b^{2} c f e^{\left (2 \, e\right )} + {\left (2 \, c f e^{\left (2 \, e\right )} + d e^{\left (2 \, e\right )}\right )} b^{3} + {\left (3 \, a^{2} b d f^{2} e^{\left (2 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (2 \, e\right )} + b^{3} d f^{2} e^{\left (2 \, e\right )}\right )} x^{2} + 2 \, {\left (3 \, {\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} a b^{2} + {\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} b^{3}\right )} x\right )} e^{\left (2 \, f x\right )}}{2 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac {{\left (3 \, a^{2} b d + b^{3} d\right )} {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )}}{f^{2}} + \frac {{\left (3 \, a^{2} b d + b^{3} d\right )} {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )}}{f^{2}} + \frac {{\left (b^{3} c f + 3 \, a b^{2} d\right )} \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac {{\left (b^{3} c f + 3 \, a b^{2} d\right )} \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} \]
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\[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\int { {\left (d x + c\right )} {\left (b \coth \left (f x + e\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (c+d x) (a+b \coth (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,\left (c+d\,x\right ) \,d x \]
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