Optimal. Leaf size=760 \[ -\frac {a (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 b (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {i a f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a^3 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^2 d^2}+\frac {i a^3 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 b f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^2 b f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^2 b f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a^2 f \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.94, antiderivative size = 760, normalized size of antiderivative = 1.00, number
of steps used = 42, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules
used = {5702, 5559, 3852, 8, 4270, 4265, 2317, 2438, 5692, 5680, 2221, 6874, 3799}
\begin {gather*} \frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a^2 b f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}-\frac {a^2 f \tanh (c+d x)}{2 b d^2 \left (a^2+b^2\right )}+\frac {a^2 b (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {a^2 b (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a^2 b (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b d \left (a^2+b^2\right )}+\frac {a^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^2 d \left (a^2+b^2\right )}+\frac {2 a^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2 \left (a^2+b^2\right )}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2 \left (a^2+b^2\right )}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 d^2 \left (a^2+b^2\right )}+\frac {a^3 (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b^2 d \left (a^2+b^2\right )}-\frac {a (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b^2 d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}-\frac {a (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b^2 d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 3852
Rule 4265
Rule 4270
Rule 5559
Rule 5680
Rule 5692
Rule 5702
Rule 6874
Rubi steps
\begin {align*} \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}-\frac {a \int (e+f x) \text {sech}^3(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {f \int \text {sech}^2(c+d x) \, dx}{2 b d}\\ &=-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}-\frac {a \int (e+f x) \text {sech}(c+d x) \, dx}{2 b^2}+\frac {a^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(i f) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 b d^2}\\ &=-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^2 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {a^2 \int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(i a f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 b^2 d}-\frac {(i a f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 b^2 d}\\ &=-\frac {a^2 b (e+f x)^2}{2 \left (a^2+b^2\right )^2 f}-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^2 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {a^3 \int (e+f x) \text {sech}^3(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}+\frac {(i a f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^2 d^2}-\frac {(i a f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^2 d^2}\\ &=-\frac {a^2 b (e+f x)^2}{2 \left (a^2+b^2\right )^2 f}-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}+\frac {a^3 \int (e+f x) \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a^2 b\right ) \int (e+f x) \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {a^3 \int (e+f x) \text {sech}(c+d x) \, dx}{2 b^2 \left (a^2+b^2\right )}-\frac {\left (a^2 b f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (a^2 b f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (a^2 f\right ) \int \text {sech}^2(c+d x) \, dx}{2 b \left (a^2+b^2\right ) d}\\ &=-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 b\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a^2 b f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (a^2 b f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (i a^2 f\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 b \left (a^2+b^2\right ) d^2}-\frac {\left (i a^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (i a^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (i a^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 b^2 \left (a^2+b^2\right ) d}+\frac {\left (i a^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 b (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a^2 f \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {\left (i a^3 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (i a^3 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (i a^3 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (i a^3 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (a^2 b f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 b (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a^2 f \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}+\frac {\left (a^2 b f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}\\ &=-\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 b (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 d^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 d^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^2 b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^2 b f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {a f \text {sech}(c+d x)}{2 b^2 d^2}+\frac {a^3 f \text {sech}(c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x) \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x)}{2 b d^2}-\frac {a^2 f \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 6.79, size = 588, normalized size = 0.77 \begin {gather*} \frac {2 a^2 b \left (-\frac {1}{2} f (c+d x)^2+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+a \left (2 a b d e (c+d x)-2 a b c f (c+d x)+a b f (c+d x)^2+2 a^2 d e \text {ArcTan}\left (e^{c+d x}\right )-2 b^2 d e \text {ArcTan}\left (e^{c+d x}\right )-2 a^2 c f \text {ArcTan}\left (e^{c+d x}\right )+2 b^2 c f \text {ArcTan}\left (e^{c+d x}\right )+i a^2 f (c+d x) \log \left (1-i e^{c+d x}\right )-i b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )-i a^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+i b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )-2 a b d e \log \left (1+e^{2 (c+d x)}\right )+2 a b c f \log \left (1+e^{2 (c+d x)}\right )-2 a b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i \left (a^2-b^2\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )+i \left (a^2-b^2\right ) f \text {PolyLog}\left (2,i e^{c+d x}\right )-a b f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )-\left (a^2+b^2\right ) d (e+f x) \text {sech}^2(c+d x) (b+a \sinh (c+d x))+\left (a^2+b^2\right ) f \text {sech}(c+d x) (-a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2067 vs. \(2 (701 ) = 1402\).
time = 4.27, size = 2068, normalized size = 2.72
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2068\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 5682 vs. \(2 (690) = 1380\).
time = 0.52, size = 5682, normalized size = 7.48 \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 (e + f x\right ) \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\mathrm {tanh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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