Integrand size = 20, antiderivative size = 401 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3} \]
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Time = 0.51 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3797, 2221, 2611, 2320, 6724, 3801, 2317, 2438, 32, 3556} \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \]
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Rule 32
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3556
Rule 3797
Rule 3801
Rule 3803
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \coth (e+f x)+3 a b^2 (c+d x)^2 \coth ^2(e+f x)+b^3 (c+d x)^2 \coth ^3(e+f x)\right ) \, dx \\ & = \frac {a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int (c+d x)^2 \coth (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \coth ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \coth ^3(e+f x) \, dx \\ & = \frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}-\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \, dx+b^3 \int (c+d x)^2 \coth (e+f x) \, dx+\frac {\left (6 a b^2 d\right ) \int (c+d x) \coth (e+f x) \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \coth ^2(e+f x) \, dx}{f} \\ & = -\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\left (2 b^3\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx+\frac {\left (b^3 d^2\right ) \int \coth (e+f x) \, dx}{f^2}-\frac {\left (6 a^2 b d\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}-\frac {\left (12 a b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \, dx}{f} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {\left (3 a^2 b d^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 a b^2 d^2\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (2 b^3 d\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {\left (3 a^2 b d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}-\frac {\left (3 a b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}-\frac {\left (b^3 d^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {\left (b^3 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3} \\ \end{align*}
Time = 7.47 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.46 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {-\frac {8 b e^{2 e} f x \left (9 a b d f (2 c+d x)+3 a^2 f^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2+3 c d f^2 x+d^2 \left (3+f^2 x^2\right )\right )\right )}{-1+e^{2 e}}+12 b \left (6 a b d f (c+d x)+3 a^2 f^2 (c+d x)^2+b^2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (1+f^2 x^2\right )\right )\right ) \log \left (1-e^{2 (e+f x)}\right )+12 b d \left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )-6 b \left (3 a^2+b^2\right ) d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )+f \text {csch}(e) \text {csch}^2(e+f x) \left (-2 b \left (9 a b f (c+d x)^2+3 a^2 f^2 x \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2 x+d^2 x \left (3+f^2 x^2\right )+3 c \left (d+d f^2 x^2\right )\right )\right ) \cosh (e)+b \left (18 a b f (c+d x)^2+3 a^2 f^2 x \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2 x+3 c d \left (2+f^2 x^2\right )+d^2 x \left (6+f^2 x^2\right )\right )\right ) \cosh (e+2 f x)+f \left (b \left (3 a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (3 e+2 f x)-2 \left (3 b^3 (c+d x)^2+a^3 f x \left (3 c^2+3 c d x+d^2 x^2\right )+3 a b^2 f x \left (3 c^2+3 c d x+d^2 x^2\right )-a \left (a^2+3 b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (2 (e+f x))\right ) \sinh (e)\right )\right )}{12 f^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1585\) vs. \(2(389)=778\).
Time = 0.63 (sec) , antiderivative size = 1586, normalized size of antiderivative = 3.96
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Leaf count of result is larger than twice the leaf count of optimal. 6356 vs. \(2 (386) = 772\).
Time = 0.34 (sec) , antiderivative size = 6356, normalized size of antiderivative = 15.85 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \]
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\[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\int \left (a + b \coth {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (386) = 772\).
Time = 0.28 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.49 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \]
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\[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \coth \left (f x + e\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \]
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