Optimal. Leaf size=721 \[ \frac {6 f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d}+\frac {6 b f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {(e+f x)^3}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.64, antiderivative size = 721, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 17, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {5569, 3720, 3716, 2190, 2531, 2282, 6589, 32, 5585, 5450, 3296, 2637, 4182, 6609, 5565, 3322, 2264} \[ -\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^2 d^3}+\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^2 d^2}+\frac {6 f^3 \sqrt {a^2+b^2} \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 f^3 \sqrt {a^2+b^2} \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^2 d^4}-\frac {6 b f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}+\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {6 b f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {3 f^2 (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^3 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {(e+f x)^3}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 3322
Rule 3716
Rule 3720
Rule 4182
Rule 5450
Rule 5565
Rule 5569
Rule 5585
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \coth ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\int (e+f x)^3 \, dx}{a}-\frac {b \int (e+f x)^3 \cosh (c+d x) \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {(3 f) \int (e+f x)^2 \coth (c+d x) \, dx}{a d}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {\int (e+f x)^3 \, dx}{a}-\frac {b \int (e+f x)^3 \text {csch}(c+d x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(6 f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a d}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {\left (2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}+\frac {\left (2 b \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2}-\frac {\left (2 b \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (3 f^3\right ) \int \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a d^3}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}-\frac {\left (3 \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {\left (3 \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}-\frac {\left (3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {\left (6 b f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a^2 d^3}-\frac {\left (6 b f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a^2 d^3}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}-\frac {\left (6 \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^2}+\frac {\left (6 \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^2}+\frac {\left (6 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}-\frac {\left (6 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^3}-\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^3}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}-\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}\\ &=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^2 d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 8.48, size = 1350, normalized size = 1.87 \[ \frac {\sqrt {a^2+b^2} \left (-2 e^3 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^3+f^3 x^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+3 e^2 f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-f^3 x^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-3 e^2 f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+3 f (e+f x)^2 \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d^2-3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d^2-6 e f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-6 f^3 x \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d+6 e f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 f^3 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 f^3 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )-6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^2 d^4}-\frac {-b d^3 x^3 \log (\cosh (c+d x)-\sinh (c+d x)+1) f^3+b d^3 x^3 \log (-\cosh (c+d x)+\sinh (c+d x)+1) f^3-3 b \left (d^2 \text {Li}_2(\cosh (c+d x)-\sinh (c+d x)) x^2+2 (d x \text {Li}_3(\cosh (c+d x)-\sinh (c+d x))+\text {Li}_4(\cosh (c+d x)-\sinh (c+d x)))\right ) f^3+3 b \left (d^2 \text {Li}_2(\sinh (c+d x)-\cosh (c+d x)) x^2+2 (d x \text {Li}_3(\sinh (c+d x)-\cosh (c+d x))+\text {Li}_4(\sinh (c+d x)-\cosh (c+d x)))\right ) f^3-3 d^2 (b d e+a f) x^2 \log (\cosh (c+d x)-\sinh (c+d x)+1) f^2+3 d^2 (b d e-a f) x^2 \log (-\cosh (c+d x)+\sinh (c+d x)+1) f^2+6 (a f-b d e) (d x \text {Li}_2(\cosh (c+d x)-\sinh (c+d x))+\text {Li}_3(\cosh (c+d x)-\sinh (c+d x))) f^2+6 (b d e+a f) (d x \text {Li}_2(\sinh (c+d x)-\cosh (c+d x))+\text {Li}_3(\sinh (c+d x)-\cosh (c+d x))) f^2-3 d^2 e (b d e+2 a f) x \log (\cosh (c+d x)-\sinh (c+d x)+1) f+3 d^2 e (b d e-2 a f) x \log (-\cosh (c+d x)+\sinh (c+d x)+1) f-3 d e (b d e-2 a f) \text {Li}_2(\cosh (c+d x)-\sinh (c+d x)) f+3 d e (b d e+2 a f) \text {Li}_2(\sinh (c+d x)-\cosh (c+d x)) f+a d^3 (e+f x)^3 (\coth (c)-1)-d^2 e^2 (b d e-3 a f) (d x-\log (-\cosh (c+d x)-\sinh (c+d x)+1))+d^2 e^2 (b d e+3 a f) (d x-\log (\cosh (c+d x)+\sinh (c+d x)+1))}{a^2 d^4}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\sinh \left (\frac {d x}{2}\right ) e^3-3 f x \sinh \left (\frac {d x}{2}\right ) e^2-3 f^2 x^2 \sinh \left (\frac {d x}{2}\right ) e-f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\sinh \left (\frac {d x}{2}\right ) e^3+3 f x \sinh \left (\frac {d x}{2}\right ) e^2+3 f^2 x^2 \sinh \left (\frac {d x}{2}\right ) e+f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.69, size = 4612, normalized size = 6.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\coth ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{3} {\left (\frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} + \frac {\sqrt {a^{2} + b^{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 )}{a^{2} d} + \frac {2}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d}\right )} - \frac {6 \, e^{2} f x}{a d} + \frac {3 \, e^{2} f \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {3 \, e^{2} f \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}} - \frac {2 \, {\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x\right )}}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} + \frac {{\left (d^{3} x^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d x + c\right )})\right )} b f^{3}}{a^{2} d^{4}} - \frac {{\left (d^{3} x^{3} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d x + c\right )})\right )} b f^{3}}{a^{2} d^{4}} + \frac {3 \, {\left (b d e^{2} f + 2 \, a e f^{2}\right )} {\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 )}}{a^{2} d^{3}} - \frac {3 \, {\left (b d e^{2} f - 2 \, a e f^{2}\right )} {\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 )}}{a^{2} d^{3}} + \frac {3 \, {\left (b d e f^{2} + a f^{3}\right )} {\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 )}}{a^{2} d^{4}} - \frac {3 \, {\left (b d e f^{2} - a f^{3}\right )} {\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 )}}{a^{2} d^{4}} - \frac {b d^{4} f^{3} x^{4} + 4 \, {\left (b d e f^{2} + a f^{3}\right )} d^{3} x^{3} + 6 \, {\left (b d^{2} e^{2} f + 2 \, a d e f^{2}\right )} d^{2} x^{2}}{4 \, a^{2} d^{4}} + \frac {b d^{4} f^{3} x^{4} + 4 \, {\left (b d e f^{2} - a f^{3}\right )} d^{3} x^{3} + 6 \, {\left (b d^{2} e^{2} f - 2 \, a d e f^{2}\right )} d^{2} x^{2}}{4 \, a^{2} d^{4}} + \int \frac {2 \, {\left ({\left (a^{2} f^{3} e^{c} + b^{2} f^{3} e^{c}\right )} x^{3} + 3 \, {\left (a^{2} e f^{2} e^{c} + b^{2} e f^{2} e^{c}\right )} x^{2} + 3 \, {\left (a^{2} e^{2} f e^{c} + b^{2} e^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}}{a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} e^{\left (d x + c\right )} - a^{2} b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{3} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________