3.2.42 \(\int \sinh ^3(c+d x) (a+b \sinh ^3(c+d x)) \, dx\) [142]

Optimal. Leaf size=99 \[ -\frac {5 b x}{16}-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {5 b \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d} \]

[Out]

-5/16*b*x-a*cosh(d*x+c)/d+1/3*a*cosh(d*x+c)^3/d+5/16*b*cosh(d*x+c)*sinh(d*x+c)/d-5/24*b*cosh(d*x+c)*sinh(d*x+c
)^3/d+1/6*b*cosh(d*x+c)*sinh(d*x+c)^5/d

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Rubi [A]
time = 0.08, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3299, 2713, 2715, 8} \begin {gather*} \frac {a \cosh ^3(c+d x)}{3 d}-\frac {a \cosh (c+d x)}{d}+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac {5 b \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {5 b \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {5 b x}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*x)/16 - (a*Cosh[c + d*x])/d + (a*Cosh[c + d*x]^3)/(3*d) + (5*b*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (5*
b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*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 ) \, dx &=i \int \left (-i a \sinh ^3(c+d x)-i b \sinh ^6(c+d x)\right ) \, dx\\ &=a \int \sinh ^3(c+d x) \, dx+b \int \sinh ^6(c+d x) \, dx\\ &=\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac {1}{6} (5 b) \int \sinh ^4(c+d x) \, dx-\frac {a \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {5 b \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d}+\frac {1}{8} (5 b) \int \sinh ^2(c+d x) \, dx\\ &=-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {5 b \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac {1}{16} (5 b) \int 1 \, dx\\ &=-\frac {5 b x}{16}-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {5 b \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 66, normalized size = 0.67 \begin {gather*} \frac {-144 a \cosh (c+d x)+16 a \cosh (3 (c+d x))+b (-60 c-60 d x+45 \sinh (2 (c+d x))-9 \sinh (4 (c+d x))+\sinh (6 (c+d x)))}{192 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-144*a*Cosh[c + d*x] + 16*a*Cosh[3*(c + d*x)] + b*(-60*c - 60*d*x + 45*Sinh[2*(c + d*x)] - 9*Sinh[4*(c + d*x)
] + Sinh[6*(c + d*x)]))/(192*d)

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Maple [A]
time = 1.31, size = 78, normalized size = 0.79

method	result	size
default	\(-\frac {5 b x}{16}-\frac {3 a \cosh \left (d x +c \right )}{4 d}+\frac {15 b \sinh \left (2 d x +2 c \right )}{64 d}-\frac {3 b \sinh \left (4 d x +4 c \right )}{64 d}+\frac {b \sinh \left (6 d x +6 c \right )}{192 d}+\frac {a \cosh \left (3 d x +3 c \right )}{12 d}\)	\(78\)
risch	\(-\frac {5 b x}{16}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}-\frac {3 b \,{\mathrm e}^{4 d x +4 c}}{128 d}+\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 d}+\frac {15 b \,{\mathrm e}^{2 d x +2 c}}{128 d}-\frac {3 a \,{\mathrm e}^{d x +c}}{8 d}-\frac {3 a \,{\mathrm e}^{-d x -c}}{8 d}-\frac {15 b \,{\mathrm e}^{-2 d x -2 c}}{128 d}+\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {3 b \,{\mathrm e}^{-4 d x -4 c}}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*b*x-3/4*a*cosh(d*x+c)/d+15/64*b*sinh(2*d*x+2*c)/d-3/64*b*sinh(4*d*x+4*c)/d+1/192*b*sinh(6*d*x+6*c)/d+1/1
2*a/d*cosh(3*d*x+3*c)

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Maxima [A]
time = 0.28, size = 143, normalized size = 1.44 \begin {gather*} -\frac {1}{384} \, 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 {\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),x, algorithm="maxima")

[Out]

-1/384*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*(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 [A]
time = 0.43, size = 135, normalized size = 1.36 \begin {gather*} \frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 8 \, a \cosh \left (d x + c\right )^{3} + 24 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 30 \, b d x - 72 \, a \cosh \left (d x + c\right ) + 3 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} + 15 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, 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),x, algorithm="fricas")

[Out]

1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 8*a*cosh(d*x + c)^3 + 24*a*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(5*b*co
sh(d*x + c)^3 - 9*b*cosh(d*x + c))*sinh(d*x + c)^3 - 30*b*d*x - 72*a*cosh(d*x + c) + 3*(b*cosh(d*x + c)^5 - 6*
b*cosh(d*x + c)^3 + 15*b*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (92) = 184\).
time = 0.44, size = 194, normalized size = 1.96 \begin {gather*} \begin {cases} \frac {a \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \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),x)

[Out]

Piecewise((a*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a*cosh(c + d*x)**3/(3*d) + 5*b*x*sinh(c + d*x)**6/16 - 15*b*
x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 15*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - 5*b*x*cosh(c + d*x)**6/
16 + 11*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*b*sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d) + 5*b*sinh(c + d
*x)*cosh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)*sinh(c)**3, True))

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Giac [A]
time = 0.44, size = 152, normalized size = 1.54 \begin {gather*} -\frac {5}{16} \, b x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {3 \, b e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {a e^{\left (3 \, d x + 3 \, c\right )}}{24 \, d} + \frac {15 \, b e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {3 \, a e^{\left (d x + c\right )}}{8 \, d} - \frac {3 \, a e^{\left (-d x - c\right )}}{8 \, d} - \frac {15 \, b e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {a e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac {3 \, b e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, 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),x, algorithm="giac")

[Out]

-5/16*b*x + 1/384*b*e^(6*d*x + 6*c)/d - 3/128*b*e^(4*d*x + 4*c)/d + 1/24*a*e^(3*d*x + 3*c)/d + 15/128*b*e^(2*d
*x + 2*c)/d - 3/8*a*e^(d*x + c)/d - 3/8*a*e^(-d*x - c)/d - 15/128*b*e^(-2*d*x - 2*c)/d + 1/24*a*e^(-3*d*x - 3*
c)/d + 3/128*b*e^(-4*d*x - 4*c)/d - 1/384*b*e^(-6*d*x - 6*c)/d

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Mupad [B]
time = 0.45, size = 67, normalized size = 0.68 \begin {gather*} \frac {\frac {a\,\mathrm {cosh}\left (3\,c+3\,d\,x\right )}{12}-\frac {3\,a\,\mathrm {cosh}\left (c+d\,x\right )}{4}+\frac {15\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{64}-\frac {3\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{64}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{192}}{d}-\frac {5\,b\,x}{16} \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),x)

[Out]

((a*cosh(3*c + 3*d*x))/12 - (3*a*cosh(c + d*x))/4 + (15*b*sinh(2*c + 2*d*x))/64 - (3*b*sinh(4*c + 4*d*x))/64 +
(b*sinh(6*c + 6*d*x))/192)/d - (5*b*x)/16

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