Optimal. Leaf size=772 \[ -\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {a^3 f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 a^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.23, antiderivative size = 772, normalized size of antiderivative = 1.00, number of steps
used = 37, number of rules used = 16, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5686, 5559,
4265, 2317, 2438, 5702, 4269, 3799, 2221, 5692, 3403, 2296, 2611, 2320, 6724, 6874}
\begin {gather*} -\frac {4 a^2 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {2 i a^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 i a^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f (e+f x) \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 )^{3/2}}-\frac {2 a^2 f (e+f x) \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 )^{3/2}}+\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b d \left (a^2+b^2\right )}-\frac {a^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3 \left (a^2+b^2\right )}-\frac {2 a^3 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 d \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2}{b^2 d \left (a^2+b^2\right )}+\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}+\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}-\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{b d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3403
Rule 3799
Rule 4265
Rule 4269
Rule 5559
Rule 5686
Rule 5692
Rule 5702
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {(2 f) \int (e+f x) \text {sech}(c+d x) \, dx}{b d}\\ &=\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(2 a f) \int (e+f x) \tanh (c+d x) \, dx}{b^2 d}-\frac {\left (2 i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (2 i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b d^2}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {\left (2 a^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2+b^2}+\frac {a^2 \int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(4 a f) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b^2 d}-\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}+\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {a^3 \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}-\frac {\left (2 a f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 d^2}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}+\frac {\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac {\left (2 a^3 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (a f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b^2 d^3}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}+\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (4 a^3 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 a^2 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 i a^2 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}-\frac {\left (2 i a^2 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}+\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 a^2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 i a^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {\left (2 i a^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {\left (2 a^3 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (a^3 f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}\\ &=-\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {a^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 9.41, size = 1183, normalized size = 1.53 \begin {gather*} \frac {a^2 \left (\frac {2 d^2 e^2 \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {2 d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {d^2 e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {d^2 e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {2 d e^c f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 d e^c f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {2 e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 a e f \text {sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c))}{\left (a^2+b^2\right ) d^2 \left (\cosh ^2(c)-\sinh ^2(c)\right )}+\frac {4 b e f \text {ArcTan}\left (\frac {\sinh (c)+\cosh (c) \tanh \left (\frac {d x}{2}\right )}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right )}{\left (a^2+b^2\right ) d^2 \sqrt {\cosh ^2(c)-\sinh ^2(c)}}+\frac {a f^2 \text {csch}(c) \left (d^2 e^{-\tanh ^{-1}(\coth (c))} x^2-\frac {i \coth (c) \left (-d x \left (-\pi +2 i \tanh ^{-1}(\coth (c))\right )-\pi \log \left (1+e^{2 d x}\right )-2 \left (i d x+i \tanh ^{-1}(\coth (c))\right ) \log \left (1-e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )+\pi \log (\cosh (d x))+2 i \tanh ^{-1}(\coth (c)) \log \left (i \sinh \left (d x+\tanh ^{-1}(\coth (c))\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )\right )}{\sqrt {1-\coth ^2(c)}}\right ) \text {sech}(c)}{\left (a^2+b^2\right ) d^3 \sqrt {\text {csch}^2(c) \left (-\cosh ^2(c)+\sinh ^2(c)\right )}}+\frac {2 b f^2 \left (-\frac {i \text {csch}(c) \left (i \left (d x+\tanh ^{-1}(\coth (c))\right ) \left (\log \left (1-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\log \left (1+e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\text {PolyLog}\left (2,e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )\right )}{\sqrt {1-\coth ^2(c)}}-\frac {2 \text {ArcTan}\left (\frac {\sinh (c)+\cosh (c) \tanh \left (\frac {d x}{2}\right )}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right ) \tanh ^{-1}(\coth (c))}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (-b e^2 \cosh (c)-2 b e f x \cosh (c)-b f^2 x^2 \cosh (c)-a e^2 \sinh (d x)-2 a e f x \sinh (d x)-a f^2 x^2 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 2.36, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\tanh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]
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. 4757 vs. \(2 (726) = 1452\).
time = 0.45, size = 4757, normalized size = 6.16 \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 )^{2} \tanh ^{2}{\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 )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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