Optimal. Leaf size=522 \[ \frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {f^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.04, antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5557, 3311, 32, 2635, 8, 3296, 2638, 3322, 2264, 2190, 2531, 2282, 6589} \[ -\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^3 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {f^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3322
Rule 5557
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac {a \int (e+f x)^2 \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {\int (e+f x)^2 \, dx}{2 b}+\frac {f^2 \int \sinh ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac {(e+f x)^3}{6 b f}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {a^2 \int (e+f x)^2 \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(2 a f) \int (e+f x) \cosh (c+d x) \, dx}{b^2 d}-\frac {f^2 \int 1 \, dx}{4 b d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac {\left (2 a^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}{b^3}-\frac {\left (2 a f^2\right ) \int \sinh (c+d x) \, dx}{b^2 d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac {\left (2 a^3\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}{b^2 \sqrt {a^2+b^2}}+\frac {\left (2 a^3\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}{b^2 \sqrt {a^2+b^2}}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^2 \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)^2 \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 {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {\left (2 a^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}{b^3 \sqrt {a^2+b^2} d}-\frac {\left (2 a^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}{b^3 \sqrt {a^2+b^2} d}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^2 \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)^2 \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 {2 a^3 f (e+f x) \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 {2 a^3 f (e+f x) \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 {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {\left (2 a^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}{b^3 \sqrt {a^2+b^2} d^2}-\frac {\left (2 a^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}{b^3 \sqrt {a^2+b^2} d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^2 \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)^2 \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 {2 a^3 f (e+f x) \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 {2 a^3 f (e+f x) \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 {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {\left (2 a^3 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 )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {\left (2 a^3 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 )}{b^3 \sqrt {a^2+b^2} d^3}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^2 \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)^2 \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 {2 a^3 f (e+f x) \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 {2 a^3 f (e+f x) \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 {2 a^3 f^2 \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 {2 a^3 f^2 \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 {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.23, size = 740, normalized size = 1.42 \[ \frac {24 a^2 e^2 x+24 a^2 e f x^2+8 a^2 f^2 x^3+\frac {48 a^3 f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )}{d^3 \sqrt {a^2+b^2}}-\frac {48 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \sqrt {a^2+b^2}}-\frac {48 a^3 f (e+f x) \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )}{d^2 \sqrt {a^2+b^2}}+\frac {48 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \sqrt {a^2+b^2}}+\frac {48 a^3 e^2 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{d \sqrt {a^2+b^2}}-\frac {48 a^3 e f x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \sqrt {a^2+b^2}}+\frac {48 a^3 e f x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \sqrt {a^2+b^2}}-\frac {24 a^3 f^2 x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \sqrt {a^2+b^2}}+\frac {24 a^3 f^2 x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \sqrt {a^2+b^2}}-\frac {48 a b f^2 \cosh (c+d x)}{d^3}+\frac {48 a b e f \sinh (c+d x)}{d^2}+\frac {48 a b f^2 x \sinh (c+d x)}{d^2}-\frac {24 a b e^2 \cosh (c+d x)}{d}-\frac {48 a b e f x \cosh (c+d x)}{d}-\frac {24 a b f^2 x^2 \cosh (c+d x)}{d}+\frac {3 b^2 f^2 \sinh (2 (c+d x))}{d^3}-\frac {6 b^2 e f \cosh (2 (c+d x))}{d^2}-\frac {6 b^2 f^2 x \cosh (2 (c+d x))}{d^2}+\frac {6 b^2 e^2 \sinh (2 (c+d x))}{d}+\frac {12 b^2 e f x \sinh (2 (c+d x))}{d}+\frac {6 b^2 f^2 x^2 \sinh (2 (c+d x))}{d}-12 b^2 e^2 x-12 b^2 e f x^2-4 b^2 f^2 x^3}{24 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.51, size = 3247, normalized size = 6.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, e^{2} {\left (\frac {8 \, a^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{3} d} + \frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{b^{2} d} - \frac {4 \, {\left (2 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{3} d} + \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2} d}\right )} + \frac {{\left (8 \, {\left (2 \, a^{2} d^{3} f^{2} e^{\left (2 \, c\right )} - b^{2} d^{3} f^{2} e^{\left (2 \, c\right )}\right )} x^{3} + 24 \, {\left (2 \, a^{2} d^{3} e f e^{\left (2 \, c\right )} - b^{2} d^{3} e f e^{\left (2 \, c\right )}\right )} x^{2} + 3 \, {\left (2 \, b^{2} d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \, {\left (2 \, d^{2} e f - d f^{2}\right )} b^{2} x e^{\left (4 \, c\right )} - {\left (2 \, d e f - f^{2}\right )} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 24 \, {\left (a b d^{2} f^{2} x^{2} e^{\left (3 \, c\right )} + 2 \, {\left (d^{2} e f - d f^{2}\right )} a b x e^{\left (3 \, c\right )} - 2 \, {\left (d e f - f^{2}\right )} a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} - 24 \, {\left (a b d^{2} f^{2} x^{2} e^{c} + 2 \, {\left (d^{2} e f + d f^{2}\right )} a b x e^{c} + 2 \, {\left (d e f + f^{2}\right )} a b e^{c}\right )} e^{\left (-d x\right )} - 3 \, {\left (2 \, b^{2} d^{2} f^{2} x^{2} + 2 \, {\left (2 \, d^{2} e f + d f^{2}\right )} b^{2} x + {\left (2 \, d e f + f^{2}\right )} b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{48 \, b^{3} d^{3}} - \int \frac {2 \, {\left (a^{3} f^{2} x^{2} e^{c} + 2 \, a^{3} e f x e^{c}\right )} e^{\left (d x\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} - b^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-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]
________________________________________________________________________________________