Optimal. Leaf size=712 \[ \frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4 \sqrt {a^2+b^2}}+\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4 \sqrt {a^2+b^2}}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}+\frac {3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f} \]
[Out]
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Rubi [A] time = 1.23, antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5557, 3311, 32, 3310, 3296, 2637, 3322, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {6 a^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {6 a^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {3 a^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {3 a^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^2 \sqrt {a^2+b^2}}-\frac {6 a^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4 \sqrt {a^2+b^2}}+\frac {6 a^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^4 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 3310
Rule 3311
Rule 3322
Rule 5557
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {a \int (e+f x)^3 \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {\int (e+f x)^3 \, dx}{2 b}+\frac {\left (3 f^2\right ) \int (e+f x) \sinh ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {a^2 \int (e+f x)^3 \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(3 a f) \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2 d}-\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\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}{b^3}-\frac {\left (6 a f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d^2}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\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}{b^2 \sqrt {a^2+b^2}}+\frac {\left (2 a^3\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}{b^2 \sqrt {a^2+b^2}}+\frac {\left (6 a f^3\right ) \int \cosh (c+d x) \, dx}{b^2 d^3}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (3 a^3 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}{b^3 \sqrt {a^2+b^2} d}-\frac {\left (3 a^3 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}{b^3 \sqrt {a^2+b^2} d}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (6 a^3 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}{b^3 \sqrt {a^2+b^2} d^2}-\frac {\left (6 a^3 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}{b^3 \sqrt {a^2+b^2} d^2}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (6 a^3 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}{b^3 \sqrt {a^2+b^2} d^3}+\frac {\left (6 a^3 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}{b^3 \sqrt {a^2+b^2} d^3}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (6 a^3 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 )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {\left (6 a^3 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 )}{b^3 \sqrt {a^2+b^2} d^4}\\ &=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {3 f^3 \sinh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \sinh ^2(c+d x)}{4 b d^2}\\ \end {align*}
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Mathematica [A] time = 4.54, size = 1407, normalized size = 1.98 \[ \frac {4 a^2 \sqrt {a^2+b^2} f^3 x^4 d^4-2 b^2 \sqrt {a^2+b^2} f^3 x^4 d^4+16 a^2 \sqrt {a^2+b^2} e f^2 x^3 d^4-8 b^2 \sqrt {a^2+b^2} e f^2 x^3 d^4+24 a^2 \sqrt {a^2+b^2} e^2 f x^2 d^4-12 b^2 \sqrt {a^2+b^2} e^2 f x^2 d^4+16 a^2 \sqrt {a^2+b^2} e^3 x d^4-8 b^2 \sqrt {a^2+b^2} e^3 x d^4+32 a^3 e^3 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^3-16 a b \sqrt {a^2+b^2} e^3 \cosh (c+d x) d^3-16 a b \sqrt {a^2+b^2} f^3 x^3 \cosh (c+d x) d^3-48 a b \sqrt {a^2+b^2} e f^2 x^2 \cosh (c+d x) d^3-48 a b \sqrt {a^2+b^2} e^2 f x \cosh (c+d x) d^3-16 a^3 f^3 x^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-48 a^3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-48 a^3 e^2 f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+16 a^3 f^3 x^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+48 a^3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+48 a^3 e^2 f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+4 b^2 \sqrt {a^2+b^2} e^3 \sinh (2 (c+d x)) d^3+4 b^2 \sqrt {a^2+b^2} f^3 x^3 \sinh (2 (c+d x)) d^3+12 b^2 \sqrt {a^2+b^2} e f^2 x^2 \sinh (2 (c+d x)) d^3+12 b^2 \sqrt {a^2+b^2} e^2 f x \sinh (2 (c+d x)) d^3-6 b^2 \sqrt {a^2+b^2} f^3 x^2 \cosh (2 (c+d x)) d^2-6 b^2 \sqrt {a^2+b^2} e^2 f \cosh (2 (c+d x)) d^2-12 b^2 \sqrt {a^2+b^2} e f^2 x \cosh (2 (c+d x)) d^2-48 a^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+48 a^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+48 a b \sqrt {a^2+b^2} f^3 x^2 \sinh (c+d x) d^2+48 a b \sqrt {a^2+b^2} e^2 f \sinh (c+d x) d^2+96 a b \sqrt {a^2+b^2} e f^2 x \sinh (c+d x) d^2-96 a b \sqrt {a^2+b^2} e f^2 \cosh (c+d x) d-96 a b \sqrt {a^2+b^2} f^3 x \cosh (c+d x) d+96 a^3 e f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d+96 a^3 f^3 x \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-96 a^3 e f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d-96 a^3 f^3 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 b^2 \sqrt {a^2+b^2} e f^2 \sinh (2 (c+d x)) d+6 b^2 \sqrt {a^2+b^2} f^3 x \sinh (2 (c+d x)) d-3 b^2 \sqrt {a^2+b^2} f^3 \cosh (2 (c+d x))-96 a^3 f^3 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+96 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+96 a b \sqrt {a^2+b^2} f^3 \sinh (c+d x)}{16 b^3 \sqrt {a^2+b^2} d^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.71, size = 5191, normalized size = 7.29 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ -\frac {1}{8} \, e^{3} {\left (\frac {8 \, a^{3} \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 )}{\sqrt {a^{2} + b^{2}} b^{3} d} + \frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{b^{2} d} - \frac {4 \, {\left (2 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{3} d} + \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2} d}\right )} + \frac {{\left (4 \, {\left (2 \, a^{2} d^{4} f^{3} e^{\left (2 \, c\right )} - b^{2} d^{4} f^{3} e^{\left (2 \, c\right )}\right )} x^{4} + 16 \, {\left (2 \, a^{2} d^{4} e f^{2} e^{\left (2 \, c\right )} - b^{2} d^{4} e f^{2} e^{\left (2 \, c\right )}\right )} x^{3} + 24 \, {\left (2 \, a^{2} d^{4} e^{2} f e^{\left (2 \, c\right )} - b^{2} d^{4} e^{2} f e^{\left (2 \, c\right )}\right )} x^{2} + {\left (4 \, b^{2} d^{3} f^{3} x^{3} e^{\left (4 \, c\right )} + 6 \, {\left (2 \, d^{3} e f^{2} - d^{2} f^{3}\right )} b^{2} x^{2} e^{\left (4 \, c\right )} + 6 \, {\left (2 \, d^{3} e^{2} f - 2 \, d^{2} e f^{2} + d f^{3}\right )} b^{2} x e^{\left (4 \, c\right )} - 3 \, {\left (2 \, d^{2} e^{2} f - 2 \, d e f^{2} + f^{3}\right )} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 16 \, {\left (a b d^{3} f^{3} x^{3} e^{\left (3 \, c\right )} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} a b x^{2} e^{\left (3 \, c\right )} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} a b x e^{\left (3 \, c\right )} - 3 \, {\left (d^{2} e^{2} f - 2 \, d e f^{2} + 2 \, f^{3}\right )} a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} - 16 \, {\left (a b d^{3} f^{3} x^{3} e^{c} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} a b x^{2} e^{c} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} a b x e^{c} + 3 \, {\left (d^{2} e^{2} f + 2 \, d e f^{2} + 2 \, f^{3}\right )} a b e^{c}\right )} e^{\left (-d x\right )} - {\left (4 \, b^{2} d^{3} f^{3} x^{3} + 6 \, {\left (2 \, d^{3} e f^{2} + d^{2} f^{3}\right )} b^{2} x^{2} + 6 \, {\left (2 \, d^{3} e^{2} f + 2 \, d^{2} e f^{2} + d f^{3}\right )} b^{2} x + 3 \, {\left (2 \, d^{2} e^{2} f + 2 \, d e f^{2} + f^{3}\right )} b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{32 \, b^{3} d^{4}} - \int \frac {2 \, {\left (a^{3} f^{3} x^{3} e^{c} + 3 \, a^{3} e f^{2} x^{2} e^{c} + 3 \, a^{3} e^{2} f x e^{c}\right )} e^{\left (d x\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} - b^{4}}\,{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 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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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