3.2.50 \(\int \sinh ^3(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\) [150]

Optimal. Leaf size=192 \[ -\frac {5}{8} a b x-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}+\frac {5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d} \]

[Out]

-5/8*a*b*x-a^2*cosh(d*x+c)/d+b^2*cosh(d*x+c)/d+1/3*a^2*cosh(d*x+c)^3/d-4/3*b^2*cosh(d*x+c)^3/d+6/5*b^2*cosh(d*
x+c)^5/d-4/7*b^2*cosh(d*x+c)^7/d+1/9*b^2*cosh(d*x+c)^9/d+5/8*a*b*cosh(d*x+c)*sinh(d*x+c)/d-5/12*a*b*cosh(d*x+c
)*sinh(d*x+c)^3/d+1/3*a*b*cosh(d*x+c)*sinh(d*x+c)^5/d

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Rubi [A]
time = 0.13, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3299, 2713, 2715, 8} \begin {gather*} \frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \cosh (c+d x)}{d}+\frac {a b \sinh ^5(c+d x) \cosh (c+d x)}{3 d}-\frac {5 a b \sinh ^3(c+d x) \cosh (c+d x)}{12 d}+\frac {5 a b \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac {5 a b x}{8}+\frac {b^2 \cosh ^9(c+d x)}{9 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a*b*x)/8 - (a^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x])/d + (a^2*Cosh[c + d*x]^3)/(3*d) - (4*b^2*Cosh[c + d
*x]^3)/(3*d) + (6*b^2*Cosh[c + d*x]^5)/(5*d) - (4*b^2*Cosh[c + d*x]^7)/(7*d) + (b^2*Cosh[c + d*x]^9)/(9*d) + (
5*a*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) - (5*a*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(12*d) + (a*b*Cosh[c + d*x]*S
inh[c + d*x]^5)/(3*d)

Rule 8

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

Rule 2713

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]

Rule 2715

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] && Integ
erQ[2*n]

Rule 3299

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]))

Rubi steps

\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=i \int \left (-i a^2 \sinh ^3(c+d x)-2 i a b \sinh ^6(c+d x)-i b^2 \sinh ^9(c+d x)\right ) \, dx\\ &=a^2 \int \sinh ^3(c+d x) \, dx+(2 a b) \int \sinh ^6(c+d x) \, dx+b^2 \int \sinh ^9(c+d x) \, dx\\ &=\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}-\frac {1}{3} (5 a b) \int \sinh ^4(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^2 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}+\frac {1}{4} (5 a b) \int \sinh ^2(c+d x) \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}+\frac {5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}-\frac {1}{8} (5 a b) \int 1 \, dx\\ &=-\frac {5}{8} a b x-\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh ^3(c+d x)}{3 d}+\frac {6 b^2 \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}+\frac {5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac {5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac {a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]
time = 0.50, size = 125, normalized size = 0.65 \begin {gather*} \frac {-1890 \left (32 a^2-21 b^2\right ) \cosh (c+d x)+420 \left (16 a^2-21 b^2\right ) \cosh (3 (c+d x))+b (2268 b \cosh (5 (c+d x))-405 b \cosh (7 (c+d x))+35 b \cosh (9 (c+d x))-840 a (60 c+60 d x-45 \sinh (2 (c+d x))+9 \sinh (4 (c+d x))-\sinh (6 (c+d x))))}{80640 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1890*(32*a^2 - 21*b^2)*Cosh[c + d*x] + 420*(16*a^2 - 21*b^2)*Cosh[3*(c + d*x)] + b*(2268*b*Cosh[5*(c + d*x)]
- 405*b*Cosh[7*(c + d*x)] + 35*b*Cosh[9*(c + d*x)] - 840*a*(60*c + 60*d*x - 45*Sinh[2*(c + d*x)] + 9*Sinh[4*(
c + d*x)] - Sinh[6*(c + d*x)])))/(80640*d)

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Maple [A]
time = 1.42, size = 152, normalized size = 0.79

method	result	size
default	\(\frac {\left (-\frac {21 b^{2}}{64}+\frac {a^{2}}{4}\right ) \cosh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {63 b^{2}}{128}-\frac {3 a^{2}}{4}\right ) \cosh \left (d x +c \right )}{d}-\frac {5 a b x}{8}+\frac {9 b^{2} \cosh \left (5 d x +5 c \right )}{320 d}-\frac {9 b^{2} \cosh \left (7 d x +7 c \right )}{1792 d}+\frac {b^{2} \cosh \left (9 d x +9 c \right )}{2304 d}+\frac {15 a b \sinh \left (2 d x +2 c \right )}{32 d}-\frac {3 a b \sinh \left (4 d x +4 c \right )}{32 d}+\frac {a b \sinh \left (6 d x +6 c \right )}{96 d}\)	\(152\)
risch	\(-\frac {5 a b x}{8}+\frac {b^{2} {\mathrm e}^{9 d x +9 c}}{4608 d}-\frac {9 b^{2} {\mathrm e}^{7 d x +7 c}}{3584 d}+\frac {a b \,{\mathrm e}^{6 d x +6 c}}{192 d}+\frac {9 b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} a b}{64 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}-\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{2}}{128 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} a b}{64 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}+\frac {63 \,{\mathrm e}^{d x +c} b^{2}}{256 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}+\frac {63 \,{\mathrm e}^{-d x -c} b^{2}}{256 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} a b}{64 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}-\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{128 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} a b}{64 d}+\frac {9 b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}-\frac {a b \,{\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {9 b^{2} {\mathrm e}^{-7 d x -7 c}}{3584 d}+\frac {b^{2} {\mathrm e}^{-9 d x -9 c}}{4608 d}\)	\(335\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-21/64*b^2+1/4*a^2)/d*cosh(3*d*x+3*c)+(63/128*b^2-3/4*a^2)/d*cosh(d*x+c)-5/8*a*b*x+9/320*b^2*cosh(5*d*x+5
*c)/d-9/1792*b^2/d*cosh(7*d*x+7*c)+1/2304*b^2/d*cosh(9*d*x+9*c)+15/32*a*b*sinh(2*d*x+2*c)/d-3/32*a*b*sinh(4*d*
x+4*c)/d+1/96*a*b*sinh(6*d*x+6*c)/d

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Maxima [A]
time = 0.29, size = 272, normalized size = 1.42 \begin {gather*} -\frac {1}{161280} \, b^{2} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac {1}{192} \, a b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/161280*b^2*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c)
- 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x
- 7*c) + 35*e^(-9*d*x - 9*c))/d) - 1/192*a*b*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d
+ 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d) + 1/24*a^2*(e^(3*d*x + 3
*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (174) = 348\).
time = 0.44, size = 355, normalized size = 1.85 \begin {gather*} \frac {35 \, b^{2} \cosh \left (d x + c\right )^{9} + 315 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{2} \cosh \left (d x + c\right )^{7} + 5040 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2268 \, b^{2} \cosh \left (d x + c\right )^{5} + 105 \, {\left (28 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 315 \, {\left (14 \, b^{2} \cosh \left (d x + c\right )^{5} - 45 \, b^{2} \cosh \left (d x + c\right )^{3} + 36 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 50400 \, a b d x + 420 \, {\left (16 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3360 \, {\left (5 \, a b \cosh \left (d x + c\right )^{3} - 9 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 315 \, {\left (4 \, b^{2} \cosh \left (d x + c\right )^{7} - 27 \, b^{2} \cosh \left (d x + c\right )^{5} + 72 \, b^{2} \cosh \left (d x + c\right )^{3} + 4 \, {\left (16 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1890 \, {\left (32 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right ) + 5040 \, {\left (a b \cosh \left (d x + c\right )^{5} - 6 \, a b \cosh \left (d x + c\right )^{3} + 15 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{80640 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80640*(35*b^2*cosh(d*x + c)^9 + 315*b^2*cosh(d*x + c)*sinh(d*x + c)^8 - 405*b^2*cosh(d*x + c)^7 + 5040*a*b*c
osh(d*x + c)*sinh(d*x + c)^5 + 2268*b^2*cosh(d*x + c)^5 + 105*(28*b^2*cosh(d*x + c)^3 - 27*b^2*cosh(d*x + c))*
sinh(d*x + c)^6 + 315*(14*b^2*cosh(d*x + c)^5 - 45*b^2*cosh(d*x + c)^3 + 36*b^2*cosh(d*x + c))*sinh(d*x + c)^4
- 50400*a*b*d*x + 420*(16*a^2 - 21*b^2)*cosh(d*x + c)^3 + 3360*(5*a*b*cosh(d*x + c)^3 - 9*a*b*cosh(d*x + c))*
sinh(d*x + c)^3 + 315*(4*b^2*cosh(d*x + c)^7 - 27*b^2*cosh(d*x + c)^5 + 72*b^2*cosh(d*x + c)^3 + 4*(16*a^2 - 2
1*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 1890*(32*a^2 - 21*b^2)*cosh(d*x + c) + 5040*(a*b*cosh(d*x + c)^5 - 6*a
*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]
time = 1.40, size = 325, normalized size = 1.69 \begin {gather*} \begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac {15 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {15 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {5 a b x \cosh ^{6}{\left (c + d x \right )}}{8} + \frac {11 a b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {5 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac {b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {64 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 b^{2} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} \sinh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**2*cosh(c + d*x)**3/(3*d) + 5*a*b*x*sinh(c + d*x)**6/8
- 15*a*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/8 + 15*a*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/8 - 5*a*b*x*cosh(c
+ d*x)**6/8 + 11*a*b*sinh(c + d*x)**5*cosh(c + d*x)/(8*d) - 5*a*b*sinh(c + d*x)**3*cosh(c + d*x)**3/(3*d) + 5
*a*b*sinh(c + d*x)*cosh(c + d*x)**5/(8*d) + b**2*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*b**2*sinh(c + d*x)**6*co
sh(c + d*x)**3/(3*d) + 16*b**2*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 64*b**2*sinh(c + d*x)**2*cosh(c + d*x
)**7/(35*d) + 128*b**2*cosh(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**2*sinh(c)**3, True))

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Giac [A]
time = 0.45, size = 301, normalized size = 1.57 \begin {gather*} -\frac {5}{8} \, a b x + \frac {b^{2} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} - \frac {9 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {a b e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} + \frac {9 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} - \frac {3 \, a b e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {15 \, a b e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} - \frac {15 \, a b e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} + \frac {3 \, a b e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac {9 \, b^{2} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} - \frac {a b e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {9 \, b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} + \frac {b^{2} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {{\left (16 \, a^{2} - 21 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {3 \, {\left (32 \, a^{2} - 21 \, b^{2}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {3 \, {\left (32 \, a^{2} - 21 \, b^{2}\right )} e^{\left (-d x - c\right )}}{256 \, d} + \frac {{\left (16 \, a^{2} - 21 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*a*b*x + 1/4608*b^2*e^(9*d*x + 9*c)/d - 9/3584*b^2*e^(7*d*x + 7*c)/d + 1/192*a*b*e^(6*d*x + 6*c)/d + 9/640
*b^2*e^(5*d*x + 5*c)/d - 3/64*a*b*e^(4*d*x + 4*c)/d + 15/64*a*b*e^(2*d*x + 2*c)/d - 15/64*a*b*e^(-2*d*x - 2*c)
/d + 3/64*a*b*e^(-4*d*x - 4*c)/d + 9/640*b^2*e^(-5*d*x - 5*c)/d - 1/192*a*b*e^(-6*d*x - 6*c)/d - 9/3584*b^2*e^
(-7*d*x - 7*c)/d + 1/4608*b^2*e^(-9*d*x - 9*c)/d + 1/384*(16*a^2 - 21*b^2)*e^(3*d*x + 3*c)/d - 3/256*(32*a^2 -
21*b^2)*e^(d*x + c)/d - 3/256*(32*a^2 - 21*b^2)*e^(-d*x - c)/d + 1/384*(16*a^2 - 21*b^2)*e^(-3*d*x - 3*c)/d

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Mupad [B]
time = 0.93, size = 149, normalized size = 0.78 \begin {gather*} \frac {\frac {a^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-a^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{3}-\frac {13\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{12}+\frac {11\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{8}-\frac {5\,d\,x\,a\,b}{8}+\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {4\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*cosh(c + d*x) - a^2*cosh(c + d*x) + (a^2*cosh(c + d*x)^3)/3 - (4*b^2*cosh(c + d*x)^3)/3 + (6*b^2*cosh(c +
d*x)^5)/5 - (4*b^2*cosh(c + d*x)^7)/7 + (b^2*cosh(c + d*x)^9)/9 - (13*a*b*cosh(c + d*x)^3*sinh(c + d*x))/12 +
(a*b*cosh(c + d*x)^5*sinh(c + d*x))/3 + (11*a*b*cosh(c + d*x)*sinh(c + d*x))/8 - (5*a*b*d*x)/8)/d

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