Optimal. Leaf size=772 \[ \frac {2 i a^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 i a^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 a^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {4 a^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b d \left (a^2+b^2\right )}-\frac {a^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3 \left (a^2+b^2\right )}-\frac {2 a^3 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 d \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2}{b^2 d \left (a^2+b^2\right )}+\frac {a f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}+\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}-\frac {a (e+f x)^2}{b^2 d}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d} \]
[Out]
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Rubi [A] time = 1.53, 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 = {5567, 5451, 4180, 2279, 2391, 5583, 4184, 3718, 2190, 5573, 3322, 2264, 2531, 2282, 6589, 6742} \[ \frac {2 a^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 a^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {a^3 f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3 \left (a^2+b^2\right )}+\frac {2 i a^2 f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 i a^2 f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 a^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {a f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {2 a^3 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2 \left (a^2+b^2\right )}-\frac {4 a^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 d \left (a^2+b^2\right )}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b d \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2}{b^2 d \left (a^2+b^2\right )}+\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}-\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3322
Rule 3718
Rule 4180
Rule 4184
Rule 5451
Rule 5567
Rule 5573
Rule 5583
Rule 6589
Rule 6742
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 8.39, size = 908, normalized size = 1.18 \[ \frac {\left (-2 e^2 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^2+f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^2+2 e f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^2-f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^2-2 e f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^2+2 f (e+f x) \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d-2 f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {f^2 \text {csch}(c) \left (d^2 e^{-\tanh ^{-1}(\coth (c))} x^2-\frac {i \coth (c) \left (-d x \left (2 i \tanh ^{-1}(\coth (c))-\pi \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 {Li}_2\left (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}{\left (a^2+b^2\right ) d^3 \sqrt {\text {csch}^2(c) \left (\sinh ^2(c)-\cosh ^2(c)\right )}}+\frac {2 e f \text {sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c)) a}{\left (a^2+b^2\right ) d^2 \left (\cosh ^2(c)-\sinh ^2(c)\right )}+\frac {2 b f^2 \left (-\frac {2 \tan ^{-1}\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)}}-\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 {Li}_2\left (-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\text {Li}_2\left (e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )\right )}{\sqrt {1-\coth ^2(c)}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (-b \cosh (c) e^2-a \sinh (d x) e^2-2 b f x \cosh (c) e-2 a f x \sinh (d x) e-b f^2 x^2 \cosh (c)-a f^2 x^2 \sinh (d x)\right )}{\left (a^2+b^2\right ) d}+\frac {4 b e f \tan ^{-1}\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)}} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.69, size = 3680, normalized size = 4.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, 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 \[ -2 \, a e f {\left (\frac {2 \, {\left (d x + c\right )}}{{\left (a^{2} + b^{2}\right )} d^{2}} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d^{2}}\right )} + 4 \, b f^{2} \int \frac {x e^{\left (d x + c\right )}}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} - 4 \, a f^{2} \int \frac {x}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} + e^{2} {\left (\frac {a^{2} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} d} - \frac {2 \, {\left (b e^{\left (-d x - c\right )} + a\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d}\right )} + \frac {4 \, b e f \arctan \left (e^{\left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} + \frac {2 \, {\left (a f^{2} x^{2} + 2 \, a e f x - {\left (b f^{2} x^{2} e^{c} + 2 \, b e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} d + b^{2} d + {\left (a^{2} d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \int -\frac {2 \, {\left (a^{2} f^{2} x^{2} e^{c} + 2 \, a^{2} e f x e^{c}\right )} e^{\left (d x\right )}}{a^{2} b + b^{3} - {\left (a^{2} b e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} e^{c} + a b^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (e + f x\right )^{2} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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