3.2.44 \(\int \sinh (c+d x) (a+b \sinh ^3(c+d x)) \, dx\) [144]

Optimal. Leaf size=60 \[ \frac {3 b x}{8}+\frac {a \cosh (c+d x)}{d}-\frac {3 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b \cosh (c+d x) \sinh ^3(c+d x)}{4 d} \]

[Out]

3/8*b*x+a*cosh(d*x+c)/d-3/8*b*cosh(d*x+c)*sinh(d*x+c)/d+1/4*b*cosh(d*x+c)*sinh(d*x+c)^3/d

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Rubi [A]
time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3299, 2718, 2715, 8} \begin {gather*} \frac {a \cosh (c+d x)}{d}+\frac {b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 b x}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*x)/8 + (a*Cosh[c + d*x])/d - (3*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^3)/
(4*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 2718

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

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

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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.75 \begin {gather*} \frac {32 a \cosh (c+d x)+b (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{32 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32*a*Cosh[c + d*x] + b*(12*(c + d*x) - 8*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(32*d)

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Maple [A]
time = 1.06, size = 47, normalized size = 0.78

method	result	size
default	\(\frac {a \cosh \left (d x +c \right )}{d}+\frac {3 b x}{8}-\frac {b \sinh \left (2 d x +2 c \right )}{4 d}+\frac {b \sinh \left (4 d x +4 c \right )}{32 d}\)	\(47\)
risch	\(\frac {3 b x}{8}+\frac {b \,{\mathrm e}^{4 d x +4 c}}{64 d}-\frac {b \,{\mathrm e}^{2 d x +2 c}}{8 d}+\frac {a \,{\mathrm e}^{d x +c}}{2 d}+\frac {a \,{\mathrm e}^{-d x -c}}{2 d}+\frac {b \,{\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {b \,{\mathrm e}^{-4 d x -4 c}}{64 d}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*cosh(d*x+c)/d+3/8*b*x-1/4*b*sinh(2*d*x+2*c)/d+1/32*b*sinh(4*d*x+4*c)/d

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Maxima [A]
time = 0.28, size = 74, normalized size = 1.23 \begin {gather*} \frac {1}{64} \, b {\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 {a \cosh \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*b*(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) + a*cosh(d
*x + c)/d

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Fricas [A]
time = 0.47, size = 63, normalized size = 1.05 \begin {gather*} \frac {b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 3 \, b d x + 8 \, a \cosh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right )^{3} - 4 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*cosh(d*x + c)*sinh(d*x + c)^3 + 3*b*d*x + 8*a*cosh(d*x + c) + (b*cosh(d*x + c)^3 - 4*b*cosh(d*x + c))*s
inh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (56) = 112\).
time = 0.19, size = 121, normalized size = 2.02 \begin {gather*} \begin {cases} \frac {a \cosh {\left (c + d x \right )}}{d} + \frac {3 b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \sinh {\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)*(a+b*sinh(d*x+c)**3),x)

[Out]

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

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Giac [A]
time = 0.40, size = 92, normalized size = 1.53 \begin {gather*} \frac {3}{8} \, b x + \frac {b e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} - \frac {b e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac {a e^{\left (d x + c\right )}}{2 \, d} + \frac {a e^{\left (-d x - c\right )}}{2 \, d} + \frac {b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac {b e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*b*x + 1/64*b*e^(4*d*x + 4*c)/d - 1/8*b*e^(2*d*x + 2*c)/d + 1/2*a*e^(d*x + c)/d + 1/2*a*e^(-d*x - c)/d + 1/
8*b*e^(-2*d*x - 2*c)/d - 1/64*b*e^(-4*d*x - 4*c)/d

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Mupad [B]
time = 0.20, size = 42, normalized size = 0.70 \begin {gather*} \frac {3\,b\,x}{8}+\frac {a\,\mathrm {cosh}\left (c+d\,x\right )-\frac {b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{32}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*b*x)/8 + (a*cosh(c + d*x) - (b*sinh(2*c + 2*d*x))/4 + (b*sinh(4*c + 4*d*x))/32)/d

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