Integrand size = 20, antiderivative size = 405 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f} \]
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Time = 0.48 (sec) , antiderivative size = 405, 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, 3799, 2221, 2611, 2320, 6724, 3801, 2317, 2438, 32, 3556} \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \]
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Rule 32
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3556
Rule 3799
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 \tanh (e+f x)+3 a b^2 (c+d x)^2 \tanh ^2(e+f x)+b^3 (c+d x)^2 \tanh ^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 \tanh (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \tanh ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \tanh ^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 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^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 \tanh (e+f x) \, dx+\frac {\left (6 a b^2 d\right ) \int (c+d x) \tanh (e+f x) \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \tanh ^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 {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 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 \tanh (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 {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 (\cosh (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) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}-\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 {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 (\cosh (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 {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}-\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 {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 (\cosh (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 (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}-\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 {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 (\cosh (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}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1163\) vs. \(2(405)=810\).
Time = 7.23 (sec) , antiderivative size = 1163, normalized size of antiderivative = 2.87 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\frac {b \left (-4 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 )+6 \left (1+e^{2 e}\right ) \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 )+6 d \left (1+e^{2 e}\right ) \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 )-3 \left (3 a^2+b^2\right ) d^2 \left (1+e^{2 e}\right ) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )\right )}{6 \left (1+e^{2 e}\right ) f^3}+\frac {\text {sech}(e) \text {sech}^2(e+f x) \left (6 b^3 c^2 f \cosh (e)+12 b^3 c d f x \cosh (e)+6 a^3 c^2 f^2 x \cosh (e)+18 a b^2 c^2 f^2 x \cosh (e)+6 b^3 d^2 f x^2 \cosh (e)+6 a^3 c d f^2 x^2 \cosh (e)+18 a b^2 c d f^2 x^2 \cosh (e)+2 a^3 d^2 f^2 x^3 \cosh (e)+6 a b^2 d^2 f^2 x^3 \cosh (e)+3 a^3 c^2 f^2 x \cosh (e+2 f x)+9 a b^2 c^2 f^2 x \cosh (e+2 f x)+3 a^3 c d f^2 x^2 \cosh (e+2 f x)+9 a b^2 c d f^2 x^2 \cosh (e+2 f x)+a^3 d^2 f^2 x^3 \cosh (e+2 f x)+3 a b^2 d^2 f^2 x^3 \cosh (e+2 f x)+3 a^3 c^2 f^2 x \cosh (3 e+2 f x)+9 a b^2 c^2 f^2 x \cosh (3 e+2 f x)+3 a^3 c d f^2 x^2 \cosh (3 e+2 f x)+9 a b^2 c d f^2 x^2 \cosh (3 e+2 f x)+a^3 d^2 f^2 x^3 \cosh (3 e+2 f x)+3 a b^2 d^2 f^2 x^3 \cosh (3 e+2 f x)+6 b^3 c d \sinh (e)+18 a b^2 c^2 f \sinh (e)+6 b^3 d^2 x \sinh (e)+36 a b^2 c d f x \sinh (e)+18 a^2 b c^2 f^2 x \sinh (e)+6 b^3 c^2 f^2 x \sinh (e)+18 a b^2 d^2 f x^2 \sinh (e)+18 a^2 b c d f^2 x^2 \sinh (e)+6 b^3 c d f^2 x^2 \sinh (e)+6 a^2 b d^2 f^2 x^3 \sinh (e)+2 b^3 d^2 f^2 x^3 \sinh (e)-6 b^3 c d \sinh (e+2 f x)-18 a b^2 c^2 f \sinh (e+2 f x)-6 b^3 d^2 x \sinh (e+2 f x)-36 a b^2 c d f x \sinh (e+2 f x)-9 a^2 b c^2 f^2 x \sinh (e+2 f x)-3 b^3 c^2 f^2 x \sinh (e+2 f x)-18 a b^2 d^2 f x^2 \sinh (e+2 f x)-9 a^2 b c d f^2 x^2 \sinh (e+2 f x)-3 b^3 c d f^2 x^2 \sinh (e+2 f x)-3 a^2 b d^2 f^2 x^3 \sinh (e+2 f x)-b^3 d^2 f^2 x^3 \sinh (e+2 f x)+9 a^2 b c^2 f^2 x \sinh (3 e+2 f x)+3 b^3 c^2 f^2 x \sinh (3 e+2 f x)+9 a^2 b c d f^2 x^2 \sinh (3 e+2 f x)+3 b^3 c d f^2 x^2 \sinh (3 e+2 f x)+3 a^2 b d^2 f^2 x^3 \sinh (3 e+2 f x)+b^3 d^2 f^2 x^3 \sinh (3 e+2 f x)\right )}{12 f^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1065\) vs. \(2(393)=786\).
Time = 0.43 (sec) , antiderivative size = 1066, normalized size of antiderivative = 2.63
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 7298, normalized size of antiderivative = 18.02 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\text {Too large to display} \]
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\[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\int \left (a + b \tanh {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (390) = 780\).
Time = 0.38 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.15 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\frac {1}{3} \, a^{3} d^{2} x^{3} + a^{3} c d x^{2} + b^{3} c^{2} {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} + a^{3} c^{2} x + \frac {3 \, a^{2} b c^{2} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {18 \, a b^{2} c^{2} f + 6 \, b^{3} c d + {\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b c d f^{2} + b^{3} c d f^{2} + 3 \, {\left (c d f^{2} + 2 \, d^{2} f\right )} a b^{2}\right )} x^{2} + 3 \, {\left (2 \, b^{3} d^{2} + 3 \, {\left (c^{2} f^{2} + 4 \, c d f\right )} a b^{2}\right )} x + {\left (9 \, a b^{2} c^{2} f^{2} x e^{\left (4 \, e\right )} + {\left (3 \, a^{2} b d^{2} f^{2} e^{\left (4 \, e\right )} + 3 \, a b^{2} d^{2} f^{2} e^{\left (4 \, e\right )} + b^{3} d^{2} f^{2} e^{\left (4 \, e\right )}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b c d f^{2} e^{\left (4 \, e\right )} + 3 \, a b^{2} c d f^{2} e^{\left (4 \, e\right )} + b^{3} c d f^{2} e^{\left (4 \, e\right )}\right )} x^{2}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left (9 \, a b^{2} c^{2} f e^{\left (2 \, e\right )} + 3 \, b^{3} c d e^{\left (2 \, e\right )} + {\left (3 \, a^{2} b d^{2} f^{2} e^{\left (2 \, e\right )} + 3 \, a b^{2} d^{2} f^{2} e^{\left (2 \, e\right )} + b^{3} d^{2} f^{2} e^{\left (2 \, e\right )}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b c d f^{2} e^{\left (2 \, e\right )} + 3 \, {\left (c d f^{2} e^{\left (2 \, e\right )} + d^{2} f e^{\left (2 \, e\right )}\right )} a b^{2} + {\left (c d f^{2} e^{\left (2 \, e\right )} + d^{2} f e^{\left (2 \, e\right )}\right )} b^{3}\right )} x^{2} + 3 \, {\left (3 \, {\left (c^{2} f^{2} e^{\left (2 \, e\right )} + 2 \, c d f e^{\left (2 \, e\right )}\right )} a b^{2} + {\left (2 \, c d f e^{\left (2 \, e\right )} + d^{2} e^{\left (2 \, e\right )}\right )} b^{3}\right )} x\right )} e^{\left (2 \, f x\right )}}{3 \, {\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 {2 \, {\left (6 \, a b^{2} c d f + b^{3} d^{2}\right )} x}{f^{2}} + \frac {{\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} {\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )}}{2 \, f^{3}} + \frac {{\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} {\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )}}{f^{3}} + \frac {{\left (6 \, a b^{2} c d f + b^{3} d^{2}\right )} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{3}} - \frac {2 \, {\left ({\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} f^{3} x^{3} + 3 \, {\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, f^{3}} \]
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\[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \tanh \left (f x + e\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \]
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