Optimal. Leaf size=642 \[ -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \text {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \text {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \text {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \text {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \text {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \text {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \text {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \text {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \]
[Out]
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Rubi [A]
time = 1.54, antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps
used = 28, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3815, 2221,
2611, 6744, 2320, 6724, 2286, 2216, 2215, 2222} \begin {gather*} \frac {3 b^2 d^2 (c+d x) \text {Li}_2\left (-\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) \text {Li}_3\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_4\left (-\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) \text {Li}_3\left (-\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 \text {Li}_2\left (-\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 \text {Li}_4\left (-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)} \end {gather*}
Antiderivative was successfully verified.
[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*} \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx &=\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 \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 \text {Li}_2\left (-\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) \text {Li}_3\left (-\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 \text {Li}_3\left (-\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 \text {Li}_2\left (-\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) \text {Li}_3\left (-\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 {\text {Li}_3\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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) \text {Li}_3\left (-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b d^3 \text {Li}_4\left (-\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) \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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) \text {Li}_3\left (-\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) \text {Li}_3\left (-\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 \text {Li}_4\left (-\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 \text {Li}_3\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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) \text {Li}_3\left (-\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) \text {Li}_3\left (-\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 \text {Li}_4\left (-\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 {\text {Li}_3\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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) \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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) \text {Li}_3\left (-\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) \text {Li}_3\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 {\text {Li}_2\left (-\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 {\text {Li}_3\left (-\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) \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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) \text {Li}_3\left (-\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) \text {Li}_3\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1783\) vs. \(2(642)=1284\).
time = 8.05, size = 1783, normalized size = 2.78 \begin {gather*} \frac {-\frac {4 b \left (12 a b c^2 d e^{2 e} f^3 x+12 b^2 c^2 d e^{2 e} f^3 x-8 a^2 c^3 e^{2 e} f^4 x-8 a b c^3 e^{2 e} f^4 x+12 a b c d^2 e^{2 e} f^3 x^2+12 b^2 c d^2 e^{2 e} f^3 x^2-12 a^2 c^2 d e^{2 e} f^4 x^2-12 a b c^2 d e^{2 e} f^4 x^2+4 a b d^3 e^{2 e} f^3 x^3+4 b^2 d^3 e^{2 e} f^3 x^3-8 a^2 c d^2 e^{2 e} f^4 x^3-8 a b c d^2 e^{2 e} f^4 x^3-2 a^2 d^3 e^{2 e} f^4 x^4-2 a b d^3 e^{2 e} f^4 x^4-6 a b c^2 d f^2 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+6 b^2 c^2 d f^2 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )-6 a b c^2 d e^{2 e} f^2 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )-6 b^2 c^2 d e^{2 e} f^2 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+4 a^2 c^3 f^3 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )-4 a b c^3 f^3 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+4 a^2 c^3 e^{2 e} f^3 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+4 a b c^3 e^{2 e} f^3 \log \left (a-b+(a+b) e^{2 (e+f x)}\right )-12 a b c d^2 f^2 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 b^2 c d^2 f^2 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-12 a b c d^2 e^{2 e} f^2 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-12 b^2 c d^2 e^{2 e} f^2 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 a^2 c^2 d f^3 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-12 a b c^2 d f^3 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 a^2 c^2 d e^{2 e} f^3 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 a b c^2 d e^{2 e} f^3 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-6 a b d^3 f^2 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+6 b^2 d^3 f^2 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-6 a b d^3 e^{2 e} f^2 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-6 b^2 d^3 e^{2 e} f^2 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 a^2 c d^2 f^3 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-12 a b c d^2 f^3 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 a^2 c d^2 e^{2 e} f^3 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+12 a b c d^2 e^{2 e} f^3 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+4 a^2 d^3 f^3 x^3 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-4 a b d^3 f^3 x^3 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+4 a^2 d^3 e^{2 e} f^3 x^3 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+4 a b d^3 e^{2 e} f^3 x^3 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+6 d \left (b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )\right ) f (c+d x) (-b d+a f (c+d x)) \text {PolyLog}\left (2,-\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-3 d^2 \left (b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )\right ) (-b d+2 a f (c+d x)) \text {PolyLog}\left (3,-\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+3 a^2 d^3 \text {PolyLog}\left (4,-\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-3 a b d^3 \text {PolyLog}\left (4,-\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+3 a^2 d^3 e^{2 e} \text {PolyLog}\left (4,-\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+3 a b d^3 e^{2 e} \text {PolyLog}\left (4,-\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )\right )}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2682\) vs.
\(2(622)=1244\).
time = 5.31, size = 2683, normalized size = 4.18
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2683\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 993, normalized size = 1.55 \begin {gather*} -\frac {6 \, b^{2} c^{2} d f x}{a^{4} f^{2} - 2 \, a^{2} b^{2} f^{2} + b^{4} f^{2}} - \frac {2 \, {\left (4 \, f^{3} x^{3} \log \left (\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}\right ) - 6 \, f x {\rm Li}_{3}(-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}) + 3 \, {\rm Li}_{4}(-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b})\right )} a b d^{3}}{3 \, {\left (a^{4} f^{4} - 2 \, a^{2} b^{2} f^{4} + b^{4} f^{4}\right )}} + \frac {3 \, b^{2} c^{2} d \log \left ({\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )} + a - b\right )}{a^{4} f^{2} - 2 \, a^{2} b^{2} f^{2} + b^{4} f^{2}} - c^{3} {\left (\frac {2 \, a b \log \left (-{\left (a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} - a - b\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} f} + \frac {2 \, b^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )}\right )} f} - \frac {f x + e}{{\left (a^{2} + 2 \, a b + b^{2}\right )} f}\right )} - \frac {3 \, {\left (2 \, a b c d^{2} f - b^{2} d^{3}\right )} {\left (2 \, f^{2} x^{2} \log \left (\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b} + 1\right ) + 2 \, f x {\rm Li}_2\left (-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}\right ) - {\rm Li}_{3}(-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b})\right )}}{2 \, {\left (a^{4} f^{4} - 2 \, a^{2} b^{2} f^{4} + b^{4} f^{4}\right )}} - \frac {3 \, {\left (a b c^{2} d f - b^{2} c d^{2}\right )} {\left (2 \, f x \log \left (\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}\right )\right )}}{a^{4} f^{3} - 2 \, a^{2} b^{2} f^{3} + b^{4} f^{3}} + \frac {a b d^{3} f^{4} x^{4} + 2 \, {\left (2 \, a b c d^{2} f - b^{2} d^{3}\right )} f^{3} x^{3} + 6 \, {\left (a b c^{2} d f^{2} - b^{2} c d^{2} f\right )} f^{2} x^{2}}{a^{4} f^{4} - 2 \, a^{2} b^{2} f^{4} + b^{4} f^{4}} + \frac {24 \, b^{2} c^{2} d x + {\left (a^{2} d^{3} f - 2 \, a b d^{3} f + b^{2} d^{3} f\right )} x^{4} + 4 \, {\left (a^{2} c d^{2} f - 2 \, a b c d^{2} f + {\left (c d^{2} f + 2 \, d^{3}\right )} b^{2}\right )} x^{3} + 6 \, {\left (a^{2} c^{2} d f - 2 \, a b c^{2} d f + {\left (c^{2} d f + 4 \, c d^{2}\right )} b^{2}\right )} x^{2} + {\left ({\left (a^{2} d^{3} f - b^{2} d^{3} f\right )} x^{4} e^{\left (2 \, e\right )} + 4 \, {\left (a^{2} c d^{2} f - b^{2} c d^{2} f\right )} x^{3} e^{\left (2 \, e\right )} + 6 \, {\left (a^{2} c^{2} d f - b^{2} c^{2} d f\right )} x^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (a^{4} f - 2 \, a^{2} b^{2} f + b^{4} f + {\left (a^{4} f + 2 \, a^{3} b f - 2 \, a b^{3} f - b^{4} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10559 vs.
\(2 (631) = 1262\).
time = 0.77, size = 10559, normalized size = 16.45 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{3}}{\left (a + b \tanh {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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