3.1.1 \(\int \sinh ^4(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [1]

Optimal. Leaf size=89 \[ \frac {1}{16} (6 a-5 b) x-\frac {(6 a-5 b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-5 b) \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3093, 2715, 8} \begin {gather*} \frac {(6 a-5 b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}-\frac {(6 a-5 b) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac {1}{16} x (6 a-5 b)+\frac {b \sinh ^5(c+d x) \cosh (c+d x)}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]

[Out]

((6*a - 5*b)*x)/16 - ((6*a - 5*b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((6*a - 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 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 3093

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos
[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e +
f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {b \cosh (c+d x) \sinh ^5(c+d x)}{6 d}+\frac {1}{6} (6 a-5 b) \int \sinh ^4(c+d x) \, dx\\ &=\frac {(6 a-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} (-6 a+5 b) \int \sinh ^2(c+d x) \, dx\\ &=-\frac {(6 a-5 b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-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} (6 a-5 b) \int 1 \, dx\\ &=\frac {1}{16} (6 a-5 b) x-\frac {(6 a-5 b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-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 = 0.08, size = 68, normalized size = 0.76 \begin {gather*} \frac {72 a c-60 b c+72 a d x-60 b d x+(-48 a+45 b) \sinh (2 (c+d x))+(6 a-9 b) \sinh (4 (c+d x))+b \sinh (6 (c+d x))}{192 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]

[Out]

(72*a*c - 60*b*c + 72*a*d*x - 60*b*d*x + (-48*a + 45*b)*Sinh[2*(c + d*x)] + (6*a - 9*b)*Sinh[4*(c + d*x)] + b*
Sinh[6*(c + d*x)])/(192*d)

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Maple [A]
time = 1.27, size = 67, normalized size = 0.75

method	result	size
default	\(\frac {\left (-\frac {3 b}{16}+\frac {a}{8}\right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {\left (\frac {15 b}{32}-\frac {a}{2}\right ) \sinh \left (2 d x +2 c \right )}{2 d}+\frac {3 a x}{8}-\frac {5 b x}{16}+\frac {b \sinh \left (6 d x +6 c \right )}{192 d}\)	\(67\)
risch	\(-\frac {5 b x}{16}+\frac {3 a x}{8}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}+\frac {{\mathrm e}^{4 d x +4 c} a}{64 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} b}{128 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} b}{128 d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} b}{128 d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}-\frac {{\mathrm e}^{-4 d x -4 c} a}{64 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/4*(-3/16*b+1/8*a)/d*sinh(4*d*x+4*c)+1/2*(15/32*b-1/2*a)*sinh(2*d*x+2*c)/d+3/8*a*x-5/16*b*x+1/192*b/d*sinh(6*
d*x+6*c)

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Maxima [A]
time = 0.27, size = 150, normalized size = 1.69 \begin {gather*} \frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/64*a*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) - 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)

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Fricas [A]
time = 0.39, size = 122, normalized size = 1.37 \begin {gather*} \frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (6 \, a - 5 \, b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} + 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (16 \, a - 15 \, b\right )} \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)^4*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b*cosh(d*x + c)^3 + 3*(2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c)^
3 + 6*(6*a - 5*b)*d*x + 3*(b*cosh(d*x + c)^5 + 2*(2*a - 3*b)*cosh(d*x + c)^3 - (16*a - 15*b)*cosh(d*x + c))*si
nh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (82) = 164\).
time = 0.45, size = 258, normalized size = 2.90 \begin {gather*} \begin {cases} \frac {3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 a \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 a \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 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 ^{2}{\left (c \right )}\right ) \sinh ^{4}{\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)**4*(a+b*sinh(d*x+c)**2),x)

[Out]

Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a*x*cosh(c + d*x)**4/8 + 5
*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*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*si
nh(c + d*x)*cosh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)*sinh(c)**4, True))

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Giac [A]
time = 0.43, size = 125, normalized size = 1.40 \begin {gather*} \frac {1}{16} \, {\left (6 \, a - 5 \, b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (2 \, a - 3 \, b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a - 15 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a - 15 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (2 \, a - 3 \, b\right )} 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)^4*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/16*(6*a - 5*b)*x + 1/384*b*e^(6*d*x + 6*c)/d + 1/128*(2*a - 3*b)*e^(4*d*x + 4*c)/d - 1/128*(16*a - 15*b)*e^(
2*d*x + 2*c)/d + 1/128*(16*a - 15*b)*e^(-2*d*x - 2*c)/d - 1/128*(2*a - 3*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 = 0.78, size = 76, normalized size = 0.85 \begin {gather*} \frac {\frac {3\,a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{2}-12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {45\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}+18\,a\,d\,x-15\,b\,d\,x}{48\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^4*(a + b*sinh(c + d*x)^2),x)

[Out]

((3*a*sinh(4*c + 4*d*x))/2 - 12*a*sinh(2*c + 2*d*x) + (45*b*sinh(2*c + 2*d*x))/4 - (9*b*sinh(4*c + 4*d*x))/4 +
(b*sinh(6*c + 6*d*x))/4 + 18*a*d*x - 15*b*d*x)/(48*d)

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