Optimal. Leaf size=795 \[ -\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {2 b^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.28, antiderivative size = 795, normalized size of antiderivative = 1.00, number of steps
used = 44, number of rules used = 23, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {5708, 2702,
327, 213, 5570, 6873, 12, 6874, 6408, 4267, 2611, 2320, 6724, 4265, 2317, 2438, 5692, 3403, 2296,
2221, 4269, 3799, 5559} \begin {gather*} -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {4 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) b^2}{a \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 b}{\left (a^2+b^2\right ) d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^2}+\frac {f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \tanh (c+d x) b}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 213
Rule 327
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 2702
Rule 3403
Rule 3799
Rule 4265
Rule 4267
Rule 4269
Rule 5559
Rule 5570
Rule 5692
Rule 5708
Rule 6408
Rule 6724
Rule 6873
Rule 6874
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b \int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int (e+f x) \left (-\frac {\tanh ^{-1}(\cosh (c+d x))}{d}+\frac {\text {sech}(c+d x)}{d}\right ) \, dx}{a}\\ &=-\frac {(e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b \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}{a \left (a^2+b^2\right )}-\frac {\left (2 b^3\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 \left (a^2+b^2\right )}-\frac {(2 f) \int \frac {(e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text {sech}(c+d x)\right )}{d} \, dx}{a}\\ &=-\frac {(e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {\left (2 b^4\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}{a \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 b^4\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}{a \left (a^2+b^2\right )^{3/2}}-\frac {b \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int (e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text {sech}(c+d x)\right ) \, dx}{a d}\\ &=-\frac {(e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(2 f) \int \left (-(e+f x) \tanh ^{-1}(\cosh (c+d x))+(e+f x) \text {sech}(c+d x)\right ) \, dx}{a d}+\frac {\left (2 b^3 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}{a \left (a^2+b^2\right )^{3/2} d}-\frac {\left (2 b^3 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}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(2 b f) \int (e+f x) \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{a \left (a^2+b^2\right ) d}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {(2 f) \int (e+f x) \tanh ^{-1}(\cosh (c+d x)) \, dx}{a d}-\frac {(2 f) \int (e+f x) \text {sech}(c+d x) \, dx}{a d}+\frac {(4 b f) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (2 b^3 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}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (2 b^3 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}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (2 i b^2 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac {\left (2 i b^2 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\int d (e+f x)^2 \text {csch}(c+d x) \, dx}{a d}+\frac {\left (2 b^3 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 )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {\left (2 b^3 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 )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {\left (2 i b^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {\left (2 i b^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {\left (2 i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (2 i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (2 b f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\int (e+f x)^2 \text {csch}(c+d x) \, dx}{a}+\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a 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 )}{a d^3}-\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (2 f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1592\) vs. \(2(795)=1590\).
time = 11.56, size = 1592, normalized size = 2.00 \begin {gather*} 4 \left (\frac {\text {csch}(c+d x) \left (d^2 e^2 \log \left (1-e^{c+d x}\right )+2 d^2 e f x \log \left (1-e^{c+d x}\right )+d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )-d^2 e^2 \log \left (1+e^{c+d x}\right )-2 d^2 e f x \log \left (1+e^{c+d x}\right )-d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )-2 d f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )+2 d f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )+2 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )-2 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )\right ) (a+b \sinh (c+d x))}{4 a d^3 (b+a \text {csch}(c+d x))}-\frac {b^3 \text {csch}(c+d x) \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 ) (a+b \sinh (c+d x))}{4 a \left (a^2+b^2\right ) d^3 (b+a \text {csch}(c+d x))}+\frac {b e f \text {csch}(c+d x) \text {sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c)) (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2 (b+a \text {csch}(c+d x)) \left (\cosh ^2(c)-\sinh ^2(c)\right )}-\frac {a 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 ) \text {csch}(c+d x) (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d^2 (b+a \text {csch}(c+d x)) \sqrt {\cosh ^2(c)-\sinh ^2(c)}}+\frac {b f^2 \text {csch}(c) \text {csch}(c+d x) \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) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d^3 (b+a \text {csch}(c+d x)) \sqrt {\text {csch}^2(c) \left (-\cosh ^2(c)+\sinh ^2(c)\right )}}-\frac {a f^2 \text {csch}(c+d x) \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 ) (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^3 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}(c) \text {sech}(c+d x) \left (a e^2 \cosh (c)+2 a e f x \cosh (c)+a f^2 x^2 \cosh (c)-b e^2 \sinh (d x)-2 b e f x \sinh (d x)-b f^2 x^2 \sinh (d x)\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d (b+a \text {csch}(c+d x))}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 2.62, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 7596 vs. \(2 (743) = 1486\).
time = 0.51, size = 7596, normalized size = 9.55 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\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.
[In]
[Out]
________________________________________________________________________________________