3.188 \(\int (a+b \sinh ^4(c+d x)) \, dx\)

Optimal. Leaf size=52 \[ a x+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8} \]

[Out]

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

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Rubi [A]  time = 0.0339566, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2635, 8} \[ a x+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sinh[c + d*x]^4,x]

[Out]

a*x + (3*b*x)/8 - (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 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]

Rubi steps

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

Mathematica [A]  time = 0.0539226, size = 49, normalized size = 0.94 \[ a x+\frac{3 b (c+d x)}{8 d}-\frac{b \sinh (2 (c+d x))}{4 d}+\frac{b \sinh (4 (c+d x))}{32 d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sinh[c + d*x]^4,x]

[Out]

a*x + (3*b*(c + d*x))/(8*d) - (b*Sinh[2*(c + d*x)])/(4*d) + (b*Sinh[4*(c + d*x)])/(32*d)

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Maple [A]  time = 0.005, size = 44, normalized size = 0.9 \begin{align*} ax+{\frac{b}{d} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16404, size = 89, normalized size = 1.71 \begin{align*} \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 )} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.61935, size = 155, normalized size = 2.98 \begin{align*} \frac{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (8 \, a + 3 \, b\right )} d x +{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.24768, size = 100, normalized size = 1.92 \begin{align*} a x + b \left (\begin{cases} \frac{3 x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \sinh ^{4}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)**4,x)

[Out]

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

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Giac [A]  time = 1.12907, size = 99, normalized size = 1.9 \begin{align*} a x + \frac{{\left (24 \, d x -{\left (18 \, e^{\left (4 \, d x + 4 \, c\right )} - 8 \, e^{\left (2 \, d x + 2 \, c\right )} + 1\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 24 \, c + e^{\left (4 \, d x + 4 \, c\right )} - 8 \, e^{\left (2 \, d x + 2 \, c\right )}\right )} b}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 1/64*(24*d*x - (18*e^(4*d*x + 4*c) - 8*e^(2*d*x + 2*c) + 1)*e^(-4*d*x - 4*c) + 24*c + e^(4*d*x + 4*c) -
8*e^(2*d*x + 2*c))*b/d