3.401 \(\int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=1443 \[ -\frac {6 f^3 \cosh (c+d x) a^4}{b^5 d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) a^4}{b^5 d^2}+\frac {(e+f x)^3 \sinh (c+d x) a^4}{b^5 d}+\frac {6 f^2 (e+f x) \sinh (c+d x) a^4}{b^5 d^3}+\frac {\left (a^2+b^2\right ) (e+f x)^4 a^3}{4 b^6 f}-\frac {(e+f x)^3 a^3}{4 b^4 d}-\frac {(e+f x)^3 \sinh ^2(c+d x) a^3}{2 b^4 d}-\frac {3 f^2 (e+f x) \sinh ^2(c+d x) a^3}{4 b^4 d^3}-\frac {3 f^3 x a^3}{8 b^4 d^3}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^2}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^4}+\frac {3 f^3 \cosh (c+d x) \sinh (c+d x) a^3}{8 b^4 d^4}+\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x) a^3}{4 b^4 d^2}-\frac {2 f^3 \cosh ^3(c+d x) a^2}{27 b^3 d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x) a^2}{3 b^3 d^2}-\frac {40 f^3 \cosh (c+d x) a^2}{9 b^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x) a^2}{b^3 d^2}+\frac {2 (e+f x)^3 \sinh (c+d x) a^2}{3 b^3 d}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x) a^2}{3 b^3 d}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x) a^2}{9 b^3 d^3}+\frac {40 f^2 (e+f x) \sinh (c+d x) a^2}{9 b^3 d^3}-\frac {(e+f x)^3 \cosh ^4(c+d x) a}{4 b^2 d}-\frac {3 f^2 (e+f x) \cosh ^4(c+d x) a}{32 b^2 d^3}+\frac {3 (e+f x)^3 a}{32 b^2 d}-\frac {9 f^2 (e+f x) \cosh ^2(c+d x) a}{32 b^2 d^3}+\frac {45 f^3 x a}{256 b^2 d^3}+\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x) a}{128 b^2 d^4}+\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x) a}{16 b^2 d^2}+\frac {45 f^3 \cosh (c+d x) \sinh (c+d x) a}{256 b^2 d^4}+\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x) a}{32 b^2 d^2}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3} \]

[Out]

-a^3*(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d-a^3*(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+c
)/(a+(a^2+b^2)^(1/2)))/b^6/d-6*a^3*(a^2+b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^4-6*a^3*(a
^2+b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^4+1/4*a^3*(a^2+b^2)*(f*x+e)^4/b^6/f-6*a^4*f^3*c
osh(d*x+c)/b^5/d^4-2/27*a^2*f^3*cosh(d*x+c)^3/b^3/d^4-1/4*a*(f*x+e)^3*cosh(d*x+c)^4/b^2/d-1/48*f*(f*x+e)^2*cos
h(3*d*x+3*c)/b/d^2-3/400*f*(f*x+e)^2*cosh(5*d*x+5*c)/b/d^2+2/3*a^2*(f*x+e)^3*sinh(d*x+c)/b^3/d-1/2*a^3*(f*x+e)
^3*sinh(d*x+c)^2/b^4/d+1/72*f^2*(f*x+e)*sinh(3*d*x+3*c)/b/d^3+3/1000*f^2*(f*x+e)*sinh(5*d*x+5*c)/b/d^3-3/8*a^3
*f^3*x/b^4/d^3-3*a^4*f*(f*x+e)^2*cosh(d*x+c)/b^5/d^2-9/32*a*f^2*(f*x+e)*cosh(d*x+c)^2/b^2/d^3-1/3*a^2*f*(f*x+e
)^2*cosh(d*x+c)^3/b^3/d^2-3/32*a*f^2*(f*x+e)*cosh(d*x+c)^4/b^2/d^3+6*a^4*f^2*(f*x+e)*sinh(d*x+c)/b^5/d^3+3/8*a
^3*f^3*cosh(d*x+c)*sinh(d*x+c)/b^4/d^4+1/3*a^2*(f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/b^3/d+3/128*a*f^3*cosh(d*x+
c)^3*sinh(d*x+c)/b^2/d^4-3/4*a^3*f^2*(f*x+e)*sinh(d*x+c)^2/b^4/d^3+3/4*a^3*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)
/b^4/d^2+2/9*a^2*f^2*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/b^3/d^3+3/16*a*f*(f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)/b^
2/d^2+9/32*a*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2-1/4*a^3*(f*x+e)^3/b^4/d-1/216*f^3*cosh(3*d*x+3*c)/b/d
^4-3/5000*f^3*cosh(5*d*x+5*c)/b/d^4+1/48*(f*x+e)^3*sinh(3*d*x+3*c)/b/d+1/80*(f*x+e)^3*sinh(5*d*x+5*c)/b/d+3/32
*a*(f*x+e)^3/b^2/d-1/8*(f*x+e)^3*sinh(d*x+c)/b/d+3/4*f^3*cosh(d*x+c)/b/d^4+45/256*a*f^3*x/b^2/d^3-40/9*a^2*f^3
*cosh(d*x+c)/b^3/d^4+3/8*f*(f*x+e)^2*cosh(d*x+c)/b/d^2-3/4*f^2*(f*x+e)*sinh(d*x+c)/b/d^3+40/9*a^2*f^2*(f*x+e)*
sinh(d*x+c)/b^3/d^3+45/256*a*f^3*cosh(d*x+c)*sinh(d*x+c)/b^2/d^4-2*a^2*f*(f*x+e)^2*cosh(d*x+c)/b^3/d^2-3*a^3*(
a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^2-3*a^3*(a^2+b^2)*f*(f*x+e)^2*polylog(
2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^2+6*a^3*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^
(1/2)))/b^6/d^3+6*a^3*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^3+a^4*(f*x+e)^3
*sinh(d*x+c)/b^5/d

________________________________________________________________________________________

Rubi [A]  time = 2.19, antiderivative size = 1443, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 18, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5579, 5448, 3296, 2638, 5447, 3311, 32, 2635, 8, 3310, 5565, 5446, 5561, 2190, 2531, 6609, 2282, 6589} \[ -\frac {6 f^3 \cosh (c+d x) a^4}{b^5 d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) a^4}{b^5 d^2}+\frac {(e+f x)^3 \sinh (c+d x) a^4}{b^5 d}+\frac {6 f^2 (e+f x) \sinh (c+d x) a^4}{b^5 d^3}+\frac {\left (a^2+b^2\right ) (e+f x)^4 a^3}{4 b^6 f}-\frac {(e+f x)^3 a^3}{4 b^4 d}-\frac {(e+f x)^3 \sinh ^2(c+d x) a^3}{2 b^4 d}-\frac {3 f^2 (e+f x) \sinh ^2(c+d x) a^3}{4 b^4 d^3}-\frac {3 f^3 x a^3}{8 b^4 d^3}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^2}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^4}+\frac {3 f^3 \cosh (c+d x) \sinh (c+d x) a^3}{8 b^4 d^4}+\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x) a^3}{4 b^4 d^2}-\frac {2 f^3 \cosh ^3(c+d x) a^2}{27 b^3 d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x) a^2}{3 b^3 d^2}-\frac {40 f^3 \cosh (c+d x) a^2}{9 b^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x) a^2}{b^3 d^2}+\frac {2 (e+f x)^3 \sinh (c+d x) a^2}{3 b^3 d}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x) a^2}{3 b^3 d}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x) a^2}{9 b^3 d^3}+\frac {40 f^2 (e+f x) \sinh (c+d x) a^2}{9 b^3 d^3}-\frac {(e+f x)^3 \cosh ^4(c+d x) a}{4 b^2 d}-\frac {3 f^2 (e+f x) \cosh ^4(c+d x) a}{32 b^2 d^3}+\frac {3 (e+f x)^3 a}{32 b^2 d}-\frac {9 f^2 (e+f x) \cosh ^2(c+d x) a}{32 b^2 d^3}+\frac {45 f^3 x a}{256 b^2 d^3}+\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x) a}{128 b^2 d^4}+\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x) a}{16 b^2 d^2}+\frac {45 f^3 \cosh (c+d x) \sinh (c+d x) a}{256 b^2 d^4}+\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x) a}{32 b^2 d^2}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

(-3*a^3*f^3*x)/(8*b^4*d^3) + (45*a*f^3*x)/(256*b^2*d^3) - (a^3*(e + f*x)^3)/(4*b^4*d) + (3*a*(e + f*x)^3)/(32*
b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^4)/(4*b^6*f) - (6*a^4*f^3*Cosh[c + d*x])/(b^5*d^4) - (40*a^2*f^3*Cosh[c +
d*x])/(9*b^3*d^4) + (3*f^3*Cosh[c + d*x])/(4*b*d^4) - (3*a^4*f*(e + f*x)^2*Cosh[c + d*x])/(b^5*d^2) - (2*a^2*f
*(e + f*x)^2*Cosh[c + d*x])/(b^3*d^2) + (3*f*(e + f*x)^2*Cosh[c + d*x])/(8*b*d^2) - (9*a*f^2*(e + f*x)*Cosh[c
+ d*x]^2)/(32*b^2*d^3) - (2*a^2*f^3*Cosh[c + d*x]^3)/(27*b^3*d^4) - (a^2*f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*b^3
*d^2) - (3*a*f^2*(e + f*x)*Cosh[c + d*x]^4)/(32*b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x]^4)/(4*b^2*d) - (f^3*Co
sh[3*c + 3*d*x])/(216*b*d^4) - (f*(e + f*x)^2*Cosh[3*c + 3*d*x])/(48*b*d^2) - (3*f^3*Cosh[5*c + 5*d*x])/(5000*
b*d^4) - (3*f*(e + f*x)^2*Cosh[5*c + 5*d*x])/(400*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x)
)/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
])/(b^6*d) - (3*a^3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2)
- (3*a^3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^2) + (6*a^3*(a
^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^3) + (6*a^3*(a^2 + b^2)*f
^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^3) - (6*a^3*(a^2 + b^2)*f^3*PolyLog[
4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^4) - (6*a^3*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/
(a + Sqrt[a^2 + b^2]))])/(b^6*d^4) + (6*a^4*f^2*(e + f*x)*Sinh[c + d*x])/(b^5*d^3) + (40*a^2*f^2*(e + f*x)*Sin
h[c + d*x])/(9*b^3*d^3) - (3*f^2*(e + f*x)*Sinh[c + d*x])/(4*b*d^3) + (a^4*(e + f*x)^3*Sinh[c + d*x])/(b^5*d)
+ (2*a^2*(e + f*x)^3*Sinh[c + d*x])/(3*b^3*d) - ((e + f*x)^3*Sinh[c + d*x])/(8*b*d) + (3*a^3*f^3*Cosh[c + d*x]
*Sinh[c + d*x])/(8*b^4*d^4) + (45*a*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*b^2*d^4) + (3*a^3*f*(e + f*x)^2*Cosh
[c + d*x]*Sinh[c + d*x])/(4*b^4*d^2) + (9*a*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(32*b^2*d^2) + (2*a^2*f
^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b^3*d^3) + (a^2*(e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b
^3*d) + (3*a*f^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*
x])/(16*b^2*d^2) - (3*a^3*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b^4*d^3) - (a^3*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^
4*d) + (f^2*(e + f*x)*Sinh[3*c + 3*d*x])/(72*b*d^3) + ((e + f*x)^3*Sinh[3*c + 3*d*x])/(48*b*d) + (3*f^2*(e + f
*x)*Sinh[5*c + 5*d*x])/(1000*b*d^3) + ((e + f*x)^3*Sinh[5*c + 5*d*x])/(80*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 5447

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c
+ d*x)^m*Cosh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5565

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5579

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {a \int (e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int \left (-\frac {1}{8} (e+f x)^3 \cosh (c+d x)+\frac {1}{16} (e+f x)^3 \cosh (3 c+3 d x)+\frac {1}{16} (e+f x)^3 \cosh (5 c+5 d x)\right ) \, dx}{b}\\ &=-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}+\frac {a^2 \int (e+f x)^3 \cosh ^3(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\int (e+f x)^3 \cosh (3 c+3 d x) \, dx}{16 b}+\frac {\int (e+f x)^3 \cosh (5 c+5 d x) \, dx}{16 b}-\frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{8 b}+\frac {(3 a f) \int (e+f x)^2 \cosh ^4(c+d x) \, dx}{4 b^2 d}\\ &=-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac {a^4 \int (e+f x)^3 \cosh (c+d x) \, dx}{b^5}-\frac {a^3 \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b^4}+\frac {\left (2 a^2\right ) \int (e+f x)^3 \cosh (c+d x) \, dx}{3 b^3}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac {(9 a f) \int (e+f x)^2 \cosh ^2(c+d x) \, dx}{16 b^2 d}-\frac {(3 f) \int (e+f x)^2 \sinh (5 c+5 d x) \, dx}{80 b d}-\frac {f \int (e+f x)^2 \sinh (3 c+3 d x) \, dx}{16 b d}+\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{8 b d}+\frac {\left (2 a^2 f^2\right ) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 b^3 d^2}+\frac {\left (3 a f^3\right ) \int \cosh ^4(c+d x) \, dx}{32 b^2 d^3}\\ &=\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac {\left (3 a^4 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^5 d}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 b^4 d}-\frac {\left (2 a^2 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^3 d}+\frac {(9 a f) \int (e+f x)^2 \, dx}{32 b^2 d}+\frac {\left (4 a^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{9 b^3 d^2}+\frac {\left (3 f^2\right ) \int (e+f x) \cosh (5 c+5 d x) \, dx}{200 b d^2}+\frac {f^2 \int (e+f x) \cosh (3 c+3 d x) \, dx}{24 b d^2}-\frac {\left (3 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{4 b d^2}+\frac {\left (9 a f^3\right ) \int \cosh ^2(c+d x) \, dx}{128 b^2 d^3}+\frac {\left (9 a f^3\right ) \int \cosh ^2(c+d x) \, dx}{32 b^2 d^3}\\ &=\frac {3 a (e+f x)^3}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac {3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {4 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \, dx}{4 b^4 d}+\frac {\left (3 a^3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d}+\frac {\left (3 a^3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d}+\frac {\left (6 a^4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^5 d^2}+\frac {\left (4 a^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^3 d^2}+\frac {\left (3 a^3 f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 b^4 d^3}-\frac {\left (4 a^2 f^3\right ) \int \sinh (c+d x) \, dx}{9 b^3 d^3}+\frac {\left (9 a f^3\right ) \int 1 \, dx}{256 b^2 d^3}+\frac {\left (9 a f^3\right ) \int 1 \, dx}{64 b^2 d^3}-\frac {\left (3 f^3\right ) \int \sinh (5 c+5 d x) \, dx}{1000 b d^3}-\frac {f^3 \int \sinh (3 c+3 d x) \, dx}{72 b d^3}+\frac {\left (3 f^3\right ) \int \sinh (c+d x) \, dx}{4 b d^3}\\ &=\frac {45 a f^3 x}{256 b^2 d^3}-\frac {a^3 (e+f x)^3}{4 b^4 d}+\frac {3 a (e+f x)^3}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac {4 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}-\frac {3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac {40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac {45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac {\left (6 a^3 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d^2}+\frac {\left (6 a^3 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d^2}-\frac {\left (6 a^4 f^3\right ) \int \sinh (c+d x) \, dx}{b^5 d^3}-\frac {\left (3 a^3 f^3\right ) \int 1 \, dx}{8 b^4 d^3}-\frac {\left (4 a^2 f^3\right ) \int \sinh (c+d x) \, dx}{b^3 d^3}\\ &=-\frac {3 a^3 f^3 x}{8 b^4 d^3}+\frac {45 a f^3 x}{256 b^2 d^3}-\frac {a^3 (e+f x)^3}{4 b^4 d}+\frac {3 a (e+f x)^3}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac {6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac {40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}-\frac {3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac {40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac {45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac {\left (6 a^3 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d^3}-\frac {\left (6 a^3 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d^3}\\ &=-\frac {3 a^3 f^3 x}{8 b^4 d^3}+\frac {45 a f^3 x}{256 b^2 d^3}-\frac {a^3 (e+f x)^3}{4 b^4 d}+\frac {3 a (e+f x)^3}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac {6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac {40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}-\frac {3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac {40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac {45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac {\left (6 a^3 \left (a^2+b^2\right ) 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^6 d^4}-\frac {\left (6 a^3 \left (a^2+b^2\right ) 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^6 d^4}\\ &=-\frac {3 a^3 f^3 x}{8 b^4 d^3}+\frac {45 a f^3 x}{256 b^2 d^3}-\frac {a^3 (e+f x)^3}{4 b^4 d}+\frac {3 a (e+f x)^3}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac {6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac {40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac {3 f^3 \cosh (c+d x)}{4 b d^4}-\frac {3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac {9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac {a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac {3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac {f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac {3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac {3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^3}-\frac {6 a^3 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^4}-\frac {6 a^3 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^4}+\frac {6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac {40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac {3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac {3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac {45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac {f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac {(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac {3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}\\ \end {align*}

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Mathematica [B]  time = 18.20, size = 5157, normalized size = 3.57 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^3*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

Result too large to show

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fricas [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]

integrate((f*x+e)^3*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*cosh(d*x + c)^3*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)

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maple [F]  time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\cosh ^{3}\left (d x +c \right )\right ) \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/960*e^3*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d*x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-
3*d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d
*x + c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x -
c) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2*c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*
b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/34560000*(8640000*(a^5*d^4*f^3*e^(5*c) + a^3
*b^2*d^4*f^3*e^(5*c))*x^4 + 34560000*(a^5*d^4*e*f^2*e^(5*c) + a^3*b^2*d^4*e*f^2*e^(5*c))*x^3 + 51840000*(a^5*d
^4*e^2*f*e^(5*c) + a^3*b^2*d^4*e^2*f*e^(5*c))*x^2 - 1728*(125*b^5*d^3*f^3*x^3*e^(10*c) + 75*(5*d^3*e*f^2 - d^2
*f^3)*b^5*x^2*e^(10*c) + 15*(25*d^3*e^2*f - 10*d^2*e*f^2 + 2*d*f^3)*b^5*x*e^(10*c) - 3*(25*d^2*e^2*f - 10*d*e*
f^2 + 2*f^3)*b^5*e^(10*c))*e^(5*d*x) + 16875*(32*a*b^4*d^3*f^3*x^3*e^(9*c) + 24*(4*d^3*e*f^2 - d^2*f^3)*a*b^4*
x^2*e^(9*c) + 12*(8*d^3*e^2*f - 4*d^2*e*f^2 + d*f^3)*a*b^4*x*e^(9*c) - 3*(8*d^2*e^2*f - 4*d*e*f^2 + f^3)*a*b^4
*e^(9*c))*e^(4*d*x) + 40000*(4*(9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a^2*b^3*e^(8*c) + (9*d^2*e^2*f - 6*d*e*f^2 +
2*f^3)*b^5*e^(8*c) - 9*(4*a^2*b^3*d^3*f^3*e^(8*c) + b^5*d^3*f^3*e^(8*c))*x^3 - 9*(4*(3*d^3*e*f^2 - d^2*f^3)*a^
2*b^3*e^(8*c) + (3*d^3*e*f^2 - d^2*f^3)*b^5*e^(8*c))*x^2 - 3*(4*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*a^2*b^3*
e^(8*c) + (9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*b^5*e^(8*c))*x)*e^(3*d*x) - 540000*(6*(2*d^2*e^2*f - 2*d*e*f^2
+ f^3)*a^3*b^2*e^(7*c) + 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a*b^4*e^(7*c) - 4*(2*a^3*b^2*d^3*f^3*e^(7*c) + a*b
^4*d^3*f^3*e^(7*c))*x^3 - 6*(2*(2*d^3*e*f^2 - d^2*f^3)*a^3*b^2*e^(7*c) + (2*d^3*e*f^2 - d^2*f^3)*a*b^4*e^(7*c)
)*x^2 - 6*(2*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a^3*b^2*e^(7*c) + (2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a*b^4*e
^(7*c))*x)*e^(2*d*x) + 2160000*(24*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^4*b*e^(6*c) + 18*(d^2*e^2*f - 2*d*e*f^2 +
2*f^3)*a^2*b^3*e^(6*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*b^5*e^(6*c) - (8*a^4*b*d^3*f^3*e^(6*c) + 6*a^2*b^3
*d^3*f^3*e^(6*c) - b^5*d^3*f^3*e^(6*c))*x^3 - 3*(8*(d^3*e*f^2 - d^2*f^3)*a^4*b*e^(6*c) + 6*(d^3*e*f^2 - d^2*f^
3)*a^2*b^3*e^(6*c) - (d^3*e*f^2 - d^2*f^3)*b^5*e^(6*c))*x^2 - 3*(8*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a^4*b*e
^(6*c) + 6*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a^2*b^3*e^(6*c) - (d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b^5*e^(6*
c))*x)*e^(d*x) + 2160000*(24*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a^4*b*e^(4*c) + 18*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3
)*a^2*b^3*e^(4*c) - 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b^5*e^(4*c) + (8*a^4*b*d^3*f^3*e^(4*c) + 6*a^2*b^3*d^3*f
^3*e^(4*c) - b^5*d^3*f^3*e^(4*c))*x^3 + 3*(8*(d^3*e*f^2 + d^2*f^3)*a^4*b*e^(4*c) + 6*(d^3*e*f^2 + d^2*f^3)*a^2
*b^3*e^(4*c) - (d^3*e*f^2 + d^2*f^3)*b^5*e^(4*c))*x^2 + 3*(8*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a^4*b*e^(4*c)
+ 6*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a^2*b^3*e^(4*c) - (d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*b^5*e^(4*c))*x)
*e^(-d*x) + 540000*(6*(2*d^2*e^2*f + 2*d*e*f^2 + f^3)*a^3*b^2*e^(3*c) + 3*(2*d^2*e^2*f + 2*d*e*f^2 + f^3)*a*b^
4*e^(3*c) + 4*(2*a^3*b^2*d^3*f^3*e^(3*c) + a*b^4*d^3*f^3*e^(3*c))*x^3 + 6*(2*(2*d^3*e*f^2 + d^2*f^3)*a^3*b^2*e
^(3*c) + (2*d^3*e*f^2 + d^2*f^3)*a*b^4*e^(3*c))*x^2 + 6*(2*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a^3*b^2*e^(3*c)
+ (2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a*b^4*e^(3*c))*x)*e^(-2*d*x) + 40000*(4*(9*d^2*e^2*f + 6*d*e*f^2 + 2*f^
3)*a^2*b^3*e^(2*c) + (9*d^2*e^2*f + 6*d*e*f^2 + 2*f^3)*b^5*e^(2*c) + 9*(4*a^2*b^3*d^3*f^3*e^(2*c) + b^5*d^3*f^
3*e^(2*c))*x^3 + 9*(4*(3*d^3*e*f^2 + d^2*f^3)*a^2*b^3*e^(2*c) + (3*d^3*e*f^2 + d^2*f^3)*b^5*e^(2*c))*x^2 + 3*(
4*(9*d^3*e^2*f + 6*d^2*e*f^2 + 2*d*f^3)*a^2*b^3*e^(2*c) + (9*d^3*e^2*f + 6*d^2*e*f^2 + 2*d*f^3)*b^5*e^(2*c))*x
)*e^(-3*d*x) + 16875*(32*a*b^4*d^3*f^3*x^3*e^c + 24*(4*d^3*e*f^2 + d^2*f^3)*a*b^4*x^2*e^c + 12*(8*d^3*e^2*f +
4*d^2*e*f^2 + d*f^3)*a*b^4*x*e^c + 3*(8*d^2*e^2*f + 4*d*e*f^2 + f^3)*a*b^4*e^c)*e^(-4*d*x) + 1728*(125*b^5*d^3
*f^3*x^3 + 75*(5*d^3*e*f^2 + d^2*f^3)*b^5*x^2 + 15*(25*d^3*e^2*f + 10*d^2*e*f^2 + 2*d*f^3)*b^5*x + 3*(25*d^2*e
^2*f + 10*d*e*f^2 + 2*f^3)*b^5)*e^(-5*d*x))*e^(-5*c)/(b^6*d^4) + integrate(-2*((a^5*b*f^3 + a^3*b^3*f^3)*x^3 +
3*(a^5*b*e*f^2 + a^3*b^3*e*f^2)*x^2 + 3*(a^5*b*e^2*f + a^3*b^3*e^2*f)*x - ((a^6*f^3*e^c + a^4*b^2*f^3*e^c)*x^
3 + 3*(a^6*e*f^2*e^c + a^4*b^2*e*f^2*e^c)*x^2 + 3*(a^6*e^2*f*e^c + a^4*b^2*e^2*f*e^c)*x)*e^(d*x))/(b^7*e^(2*d*
x + 2*c) + 2*a*b^6*e^(d*x + c) - b^7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)

[Out]

int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)

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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.

[In]

integrate((f*x+e)**3*cosh(d*x+c)**3*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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