Integrand size = 20, antiderivative size = 642 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=-\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \]
[Out]
Time = 1.48 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3815, 2221, 2611, 6744, 2320, 6724, 2286, 2216, 2215, 2222} \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f \left (a^2-b^2\right )^2}-\frac {2 b^2 (c+d x)^3}{f \left (a^2-b^2\right )^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3}{f (a-b) (a+b)^2 \left ((a+b) e^{2 e+2 f x}+a-b\right )}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}-\frac {2 b (c+d x)^3 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f (a-b)^2 (a+b)}+\frac {(c+d x)^4}{4 d (a-b)^2}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)} \]
[In]
[Out]
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2286
Rule 2320
Rule 2611
Rule 3815
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(c+d x)^3}{(a-b)^2}+\frac {4 b e^{2 e+2 f x} (c+d x)^3}{(a-b)^2 \left (-a \left (1-\frac {b}{a}\right )-a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}\right )}+\frac {4 b^2 e^{4 e+4 f x} (c+d x)^3}{(a-b)^2 \left (a \left (1-\frac {b}{a}\right )+a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}\right )^2}\right ) \, dx \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {(4 b) \int \frac {e^{2 e+2 f x} (c+d x)^3}{-a \left (1-\frac {b}{a}\right )-a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}} \, dx}{(a-b)^2}+\frac {\left (4 b^2\right ) \int \frac {e^{4 e+4 f x} (c+d x)^3}{\left (a \left (1-\frac {b}{a}\right )+a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}\right )^2} \, dx}{(a-b)^2} \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {\left (4 b^2\right ) \int \left (\frac {(c+d x)^3}{(a+b)^2}+\frac {(a-b)^2 (c+d x)^3}{(a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right )^2}+\frac {2 (-a+b) (c+d x)^3}{(a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right )}\right ) \, dx}{(a-b)^2}+\frac {(6 b d) \int (c+d x)^2 \log \left (1+\frac {\left (1+\frac {b}{a}\right ) e^{2 e+2 f x}}{1-\frac {b}{a}}\right ) \, dx}{(a-b)^2 (a+b) f} \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {b^2 (c+d x)^4}{\left (a^2-b^2\right )^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {\left (4 b^2\right ) \int \frac {(c+d x)^3}{\left (a-b+(a+b) e^{2 e+2 f x}\right )^2} \, dx}{(a+b)^2}-\frac {\left (8 b^2\right ) \int \frac {(c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2}+\frac {\left (6 b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {\left (1+\frac {b}{a}\right ) e^{2 e+2 f x}}{1-\frac {b}{a}}\right ) \, dx}{(a-b)^2 (a+b) f^2} \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2-b^2\right )^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}+\frac {\left (4 b^2\right ) \int \frac {(c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2}+\frac {\left (8 b^2\right ) \int \frac {e^{2 e+2 f x} (c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b)}-\frac {\left (4 b^2\right ) \int \frac {e^{2 e+2 f x} (c+d x)^3}{\left (a-b+(a+b) e^{2 e+2 f x}\right )^2} \, dx}{a^2-b^2}-\frac {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {\left (1+\frac {b}{a}\right ) e^{2 e+2 f x}}{1-\frac {b}{a}}\right ) \, dx}{(a-b)^2 (a+b) f^3} \\ & = \frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {4 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {\left (4 b^2\right ) \int \frac {e^{2 e+2 f x} (c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b)}-\frac {\left (3 b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{2 (a-b)^2 (a+b) f^4}-\frac {\left (6 b^2 d\right ) \int \frac {(c+d x)^2}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2 f}-\frac {\left (12 b^2 d\right ) \int (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {6 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}-\frac {\left (12 b^2 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^2}+\frac {\left (6 b^2 d\right ) \int \frac {e^{2 e+2 f x} (c+d x)^2}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b) f}+\frac {\left (6 b^2 d\right ) \int (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {6 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {\left (6 b^2 d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^3}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^2}+\frac {\left (6 b^2 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^2} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{\left (a^2-b^2\right )^2 f^4}-\frac {\left (3 b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^3}-\frac {\left (3 b^2 d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^3} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^4}-\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{2 \left (a^2-b^2\right )^2 f^4}-\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{2 \left (a^2-b^2\right )^2 f^4} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \\ \end{align*}
Time = 5.37 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\frac {16 b c^2 f^3 (-3 b d+2 a c f) x+\frac {16 (a-b) b^2 f^3 (c+d x)^3}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )}-\frac {8 a (a-b) b f^4 (c+d x)^4}{d \left (b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )\right )}+48 b c d f^2 (b d-a c f) x \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+24 b d^2 f^2 (b d-2 a c f) x^2 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )-16 a b d^3 f^3 x^3 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+8 b c^2 f^2 (3 b d-2 a c f) \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+24 b c d f (-b d+a c f) \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )-12 b d^2 (b d-2 a c f) \left (2 f x \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+12 a b d^3 \left (2 f^2 x^2 \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+2 f x \operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (4,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+\frac {(a-b) (a+b) f^3 \left (\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)+\left (a^2-b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+2 b \left (-4 b (c+d x)^3+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \sinh (f x)\right )}{(a \cosh (e)+b \sinh (e)) (a \cosh (e+f x)+b \sinh (e+f x))}}{8 (a-b)^2 (a+b)^2 f^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2682\) vs. \(2(622)=1244\).
Time = 0.38 (sec) , antiderivative size = 2683, normalized size of antiderivative = 4.18
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Leaf count of result is larger than twice the leaf count of optimal. 6160 vs. \(2 (619) = 1238\).
Time = 0.38 (sec) , antiderivative size = 6160, normalized size of antiderivative = 9.60 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (a + b \tanh {\left (e + f x \right )}\right )^{2}}\, dx \]
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Time = 0.56 (sec) , antiderivative size = 1060, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2} \,d x \]
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