Optimal. Leaf size=734 \[ \frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}-\frac {b^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a d^3 \left (a^2+b^2\right )}-\frac {2 i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac {2 i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )}-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d} \]
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Rubi [A] time = 1.10, antiderivative size = 734, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5589, 5461, 4182, 2531, 2282, 6589, 5573, 5561, 2190, 6742, 4180, 3718} \[ -\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}-\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {b^2 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {2 i b f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac {2 i b f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a d^3 \left (a^2+b^2\right )}-\frac {b^2 f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a d^3 \left (a^2+b^2\right )}-\frac {2 i b f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac {2 i b f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )}-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 4180
Rule 4182
Rule 5461
Rule 5561
Rule 5573
Rule 5589
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a}-\frac {b \int (e+f x)^2 \text {sech}(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 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {b^2 (e+f x)^3}{3 a \left (a^2+b^2\right ) f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}\\ &=\frac {b^2 (e+f x)^3}{3 a \left (a^2+b^2\right ) f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {b \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^2 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac {f^2 \int \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}-\frac {f^2 \int \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}-\frac {2 b^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {\left (2 b^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}+\frac {(2 i b f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {(2 i b f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}+\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (2 b^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 )}{a \left (a^2+b^2\right ) d^3}+\frac {\left (2 b^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 )}{a \left (a^2+b^2\right ) d^3}-\frac {\left (2 i b f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (2 i b f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {2 b^2 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 ) d^3}+\frac {2 b^2 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 ) d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {\left (2 i b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\left (2 i b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {\left (b^2 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^2 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 ) d^3}+\frac {2 b^2 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 ) d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}\\ &=-\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (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 ) d}-\frac {b^2 (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 ) d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^2 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 ) d^2}-\frac {2 b^2 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 ) d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^2 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 ) d^3}+\frac {2 b^2 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 ) d^3}-\frac {b^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}\\ \end {align*}
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Mathematica [B] time = 28.71, size = 3002, normalized size = 4.09 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.52, size = 1539, normalized size = 2.10 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ -e^{2} {\left (\frac {b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{3} + a b^{2}\right )} d} - \frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} + \frac {2 \, {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac {2 \, {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac {{\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac {{\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} - \frac {2 \, {\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2}\right )}}{3 \, a d^{3}} + \int \frac {2 \, {\left (b^{3} f^{2} x^{2} + 2 \, b^{3} e f x - {\left (a b^{2} f^{2} x^{2} e^{c} + 2 \, a b^{2} e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{3} b + a b^{3} - {\left (a^{3} b e^{\left (2 \, c\right )} + a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{4} e^{c} + a^{2} b^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int -\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} + b^{2} + {\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, 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 \[ \int \frac {{\left (e+f\,x\right )}^2}{\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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