3.161 \(\int \sinh ^2(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)

Optimal. Leaf size=291 \[ \frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a^3 x}{2}+\frac{3 a b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{16 d}+\frac{35 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{64 d}-\frac{105 a b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{105}{128} a b^2 x+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d} \]

[Out]

-(a^3*x)/2 + (105*a*b^2*x)/128 + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d - (2*a^2*b*Cosh[c + d*x]^3)
/d + (5*b^3*Cosh[c + d*x]^3)/(3*d) + (3*a^2*b*Cosh[c + d*x]^5)/(5*d) - (2*b^3*Cosh[c + d*x]^5)/d + (10*b^3*Cos
h[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]^11)/(11*d) + (a^3*Cosh[c + d*x]*Sinh[
c + d*x])/(2*d) - (105*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(
64*d) - (7*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(16*d) + (3*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)

________________________________________________________________________________________

Rubi [A]  time = 0.208646, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3220, 2635, 8, 2633} \[ \frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a^3 x}{2}+\frac{3 a b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{16 d}+\frac{35 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{64 d}-\frac{105 a b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{105}{128} a b^2 x+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^3,x]

[Out]

-(a^3*x)/2 + (105*a*b^2*x)/128 + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d - (2*a^2*b*Cosh[c + d*x]^3)
/d + (5*b^3*Cosh[c + d*x]^3)/(3*d) + (3*a^2*b*Cosh[c + d*x]^5)/(5*d) - (2*b^3*Cosh[c + d*x]^5)/d + (10*b^3*Cos
h[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]^11)/(11*d) + (a^3*Cosh[c + d*x]*Sinh[
c + d*x])/(2*d) - (105*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(
64*d) - (7*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(16*d) + (3*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

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 8

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\int \left (-a^3 \sinh ^2(c+d x)-3 a^2 b \sinh ^5(c+d x)-3 a b^2 \sinh ^8(c+d x)-b^3 \sinh ^{11}(c+d x)\right ) \, dx\\ &=a^3 \int \sinh ^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh ^5(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^8(c+d x) \, dx+b^3 \int \sinh ^{11}(c+d x) \, dx\\ &=\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}-\frac{1}{2} a^3 \int 1 \, dx-\frac{1}{8} \left (21 a b^2\right ) \int \sinh ^6(c+d x) \, dx+\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}-\frac{b^3 \operatorname{Subst}\left (\int \left (1-5 x^2+10 x^4-10 x^6+5 x^8-x^{10}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^3 x}{2}+\frac{3 a^2 b \cosh (c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}+\frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}+\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{7 a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{16 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}+\frac{1}{16} \left (35 a b^2\right ) \int \sinh ^4(c+d x) \, dx\\ &=-\frac{a^3 x}{2}+\frac{3 a^2 b \cosh (c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}+\frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}+\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{35 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{64 d}-\frac{7 a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{16 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}-\frac{1}{64} \left (105 a b^2\right ) \int \sinh ^2(c+d x) \, dx\\ &=-\frac{a^3 x}{2}+\frac{3 a^2 b \cosh (c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}+\frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}+\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{105 a b^2 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{35 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{64 d}-\frac{7 a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{16 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}+\frac{1}{128} \left (105 a b^2\right ) \int 1 \, dx\\ &=-\frac{a^3 x}{2}+\frac{105}{128} a b^2 x+\frac{3 a^2 b \cosh (c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}+\frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}+\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{105 a b^2 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{35 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{64 d}-\frac{7 a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{16 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.476869, size = 194, normalized size = 0.67 \[ \frac{-27720 a \left (64 a^2-105 b^2\right ) (c+d x)+110880 a \left (8 a^2-21 b^2\right ) \sinh (2 (c+d x))-20790 b \left (77 b^2-320 a^2\right ) \cosh (c+d x)+34650 b \left (11 b^2-32 a^2\right ) \cosh (3 (c+d x))-2079 b \left (55 b^2-64 a^2\right ) \cosh (5 (c+d x))+582120 a b^2 \sinh (4 (c+d x))-110880 a b^2 \sinh (6 (c+d x))+10395 a b^2 \sinh (8 (c+d x))+27225 b^3 \cosh (7 (c+d x))-4235 b^3 \cosh (9 (c+d x))+315 b^3 \cosh (11 (c+d x))}{3548160 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^3,x]

[Out]

(-27720*a*(64*a^2 - 105*b^2)*(c + d*x) - 20790*b*(-320*a^2 + 77*b^2)*Cosh[c + d*x] + 34650*b*(-32*a^2 + 11*b^2
)*Cosh[3*(c + d*x)] - 2079*b*(-64*a^2 + 55*b^2)*Cosh[5*(c + d*x)] + 27225*b^3*Cosh[7*(c + d*x)] - 4235*b^3*Cos
h[9*(c + d*x)] + 315*b^3*Cosh[11*(c + d*x)] + 110880*a*(8*a^2 - 21*b^2)*Sinh[2*(c + d*x)] + 582120*a*b^2*Sinh[
4*(c + d*x)] - 110880*a*b^2*Sinh[6*(c + d*x)] + 10395*a*b^2*Sinh[8*(c + d*x)])/(3548160*d)

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 188, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( -{\frac{256}{693}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{11}}-{\frac{10\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{99}}+{\frac{80\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{693}}-{\frac{32\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{231}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{693}} \right ) \cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ( \left ( 1/8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}-{\frac{7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{48}}+{\frac{35\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{192}}-{\frac{35\,\sinh \left ( dx+c \right ) }{128}} \right ) \cosh \left ( dx+c \right ) +{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) +3\,{a}^{2}b \left ({\frac{8}{15}}+1/5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +{a}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^3)^3,x)

[Out]

1/d*(b^3*(-256/693+1/11*sinh(d*x+c)^10-10/99*sinh(d*x+c)^8+80/693*sinh(d*x+c)^6-32/231*sinh(d*x+c)^4+128/693*s
inh(d*x+c)^2)*cosh(d*x+c)+3*a*b^2*((1/8*sinh(d*x+c)^7-7/48*sinh(d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*sinh(d*x+
c))*cosh(d*x+c)+35/128*d*x+35/128*c)+3*a^2*b*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+a^3*(1/2*
cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c))

________________________________________________________________________________________

Maxima [A]  time = 1.09023, size = 522, normalized size = 1.79 \begin{align*} -\frac{1}{8} \, a^{3}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{1419264} \, b^{3}{\left (\frac{{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac{320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac{1}{2048} \, a b^{2}{\left (\frac{{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{1680 \,{\left (d x + c\right )}}{d} - \frac{672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac{1}{160} \, a^{2} b{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-1/8*a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/1419264*b^3*((847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*
x - 4*c) + 22869*e^(-6*d*x - 6*c) - 76230*e^(-8*d*x - 8*c) + 320166*e^(-10*d*x - 10*c) - 63)*e^(11*d*x + 11*c)
/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x - 3*c) + 22869*e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-
9*d*x - 9*c) - 63*e^(-11*d*x - 11*c))/d) - 1/2048*a*b^2*((32*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 672*e^(
-6*d*x - 6*c) - 3)*e^(8*d*x + 8*c)/d - 1680*(d*x + c)/d - (672*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 32*e^
(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*c))/d) + 1/160*a^2*b*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x
+ c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d)

________________________________________________________________________________________

Fricas [B]  time = 1.93065, size = 1546, normalized size = 5.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

1/3548160*(315*b^3*cosh(d*x + c)^11 + 3465*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - 4235*b^3*cosh(d*x + c)^9 + 831
60*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 27225*b^3*cosh(d*x + c)^7 + 3465*(15*b^3*cosh(d*x + c)^3 - 11*b^3*cos
h(d*x + c))*sinh(d*x + c)^8 + 1155*(126*b^3*cosh(d*x + c)^5 - 308*b^3*cosh(d*x + c)^3 + 165*b^3*cosh(d*x + c))
*sinh(d*x + c)^6 + 2079*(64*a^2*b - 55*b^3)*cosh(d*x + c)^5 + 83160*(7*a*b^2*cosh(d*x + c)^3 - 8*a*b^2*cosh(d*
x + c))*sinh(d*x + c)^5 + 3465*(30*b^3*cosh(d*x + c)^7 - 154*b^3*cosh(d*x + c)^5 + 275*b^3*cosh(d*x + c)^3 + 3
*(64*a^2*b - 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 34650*(32*a^2*b - 11*b^3)*cosh(d*x + c)^3 + 27720*(21*a*
b^2*cosh(d*x + c)^5 - 80*a*b^2*cosh(d*x + c)^3 + 84*a*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - 27720*(64*a^3 - 105
*a*b^2)*d*x + 3465*(5*b^3*cosh(d*x + c)^9 - 44*b^3*cosh(d*x + c)^7 + 165*b^3*cosh(d*x + c)^5 + 6*(64*a^2*b - 5
5*b^3)*cosh(d*x + c)^3 - 30*(32*a^2*b - 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 20790*(320*a^2*b - 77*b^3)*co
sh(d*x + c) + 27720*(3*a*b^2*cosh(d*x + c)^7 - 24*a*b^2*cosh(d*x + c)^5 + 84*a*b^2*cosh(d*x + c)^3 + 8*(8*a^3
- 21*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 83.4585, size = 498, normalized size = 1.71 \begin{align*} \begin{cases} \frac{a^{3} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{a^{3} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{a^{3} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} + \frac{3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac{105 a b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac{105 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac{315 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac{105 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac{105 a b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac{279 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{128 d} - \frac{511 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac{385 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac{105 a b^{2} \sinh{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac{b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{10 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{3 d} - \frac{32 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac{128 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{63 d} - \frac{256 b^{3} \cosh ^{11}{\left (c + d x \right )}}{693 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{3} \sinh ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**2*(a+b*sinh(d*x+c)**3)**3,x)

[Out]

Piecewise((a**3*x*sinh(c + d*x)**2/2 - a**3*x*cosh(c + d*x)**2/2 + a**3*sinh(c + d*x)*cosh(c + d*x)/(2*d) + 3*
a**2*b*sinh(c + d*x)**4*cosh(c + d*x)/d - 4*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**3/d + 8*a**2*b*cosh(c + d*x
)**5/(5*d) + 105*a*b**2*x*sinh(c + d*x)**8/128 - 105*a*b**2*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 315*a*b**
2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 105*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 105*a*b**2*x*co
sh(c + d*x)**8/128 + 279*a*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511*a*b**2*sinh(c + d*x)**5*cosh(c +
d*x)**3/(128*d) + 385*a*b**2*sinh(c + d*x)**3*cosh(c + d*x)**5/(128*d) - 105*a*b**2*sinh(c + d*x)*cosh(c + d*x
)**7/(128*d) + b**3*sinh(c + d*x)**10*cosh(c + d*x)/d - 10*b**3*sinh(c + d*x)**8*cosh(c + d*x)**3/(3*d) + 16*b
**3*sinh(c + d*x)**6*cosh(c + d*x)**5/(3*d) - 32*b**3*sinh(c + d*x)**4*cosh(c + d*x)**7/(7*d) + 128*b**3*sinh(
c + d*x)**2*cosh(c + d*x)**9/(63*d) - 256*b**3*cosh(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**3*
sinh(c)**2, True))

________________________________________________________________________________________

Giac [A]  time = 1.44249, size = 545, normalized size = 1.87 \begin{align*} \frac{315 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} - 4235 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 10395 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 27225 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 110880 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 133056 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 114345 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 582120 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1108800 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 381150 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 887040 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 2328480 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6652800 \, a^{2} b e^{\left (d x + c\right )} - 1600830 \, b^{3} e^{\left (d x + c\right )} - 55440 \,{\left (64 \, a^{3} - 105 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (582120 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 110880 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 27225 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 10395 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 4235 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, b^{3} - 20790 \,{\left (320 \, a^{2} b - 77 \, b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )} + 110880 \,{\left (8 \, a^{3} - 21 \, a b^{2}\right )} e^{\left (9 \, d x + 9 \, c\right )} + 34650 \,{\left (32 \, a^{2} b - 11 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} - 2079 \,{\left (64 \, a^{2} b - 55 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{7096320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^3)^3,x, algorithm="giac")

[Out]

1/7096320*(315*b^3*e^(11*d*x + 11*c) - 4235*b^3*e^(9*d*x + 9*c) + 10395*a*b^2*e^(8*d*x + 8*c) + 27225*b^3*e^(7
*d*x + 7*c) - 110880*a*b^2*e^(6*d*x + 6*c) + 133056*a^2*b*e^(5*d*x + 5*c) - 114345*b^3*e^(5*d*x + 5*c) + 58212
0*a*b^2*e^(4*d*x + 4*c) - 1108800*a^2*b*e^(3*d*x + 3*c) + 381150*b^3*e^(3*d*x + 3*c) + 887040*a^3*e^(2*d*x + 2
*c) - 2328480*a*b^2*e^(2*d*x + 2*c) + 6652800*a^2*b*e^(d*x + c) - 1600830*b^3*e^(d*x + c) - 55440*(64*a^3 - 10
5*a*b^2)*(d*x + c) - (582120*a*b^2*e^(7*d*x + 7*c) - 110880*a*b^2*e^(5*d*x + 5*c) - 27225*b^3*e^(4*d*x + 4*c)
+ 10395*a*b^2*e^(3*d*x + 3*c) + 4235*b^3*e^(2*d*x + 2*c) - 315*b^3 - 20790*(320*a^2*b - 77*b^3)*e^(10*d*x + 10
*c) + 110880*(8*a^3 - 21*a*b^2)*e^(9*d*x + 9*c) + 34650*(32*a^2*b - 11*b^3)*e^(8*d*x + 8*c) - 2079*(64*a^2*b -
55*b^3)*e^(6*d*x + 6*c))*e^(-11*d*x - 11*c))/d