3.367 \(\int (x+\sinh (x))^3 \, dx\)

Optimal. Leaf size=56 \[ \frac{x^4}{4}-\frac{3 x^2}{4}+3 x^2 \cosh (x)-\frac{3 \sinh ^2(x)}{4}-6 x \sinh (x)+\frac{\cosh ^3(x)}{3}+5 \cosh (x)+\frac{3}{2} x \sinh (x) \cosh (x) \]

[Out]

(-3*x^2)/4 + x^4/4 + 5*Cosh[x] + 3*x^2*Cosh[x] + Cosh[x]^3/3 - 6*x*Sinh[x] + (3*x*Cosh[x]*Sinh[x])/2 - (3*Sinh
[x]^2)/4

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Rubi [A]  time = 0.076909, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6742, 3296, 2638, 3310, 30, 2633} \[ \frac{x^4}{4}-\frac{3 x^2}{4}+3 x^2 \cosh (x)-\frac{3 \sinh ^2(x)}{4}-6 x \sinh (x)+\frac{\cosh ^3(x)}{3}+5 \cosh (x)+\frac{3}{2} x \sinh (x) \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[(x + Sinh[x])^3,x]

[Out]

(-3*x^2)/4 + x^4/4 + 5*Cosh[x] + 3*x^2*Cosh[x] + Cosh[x]^3/3 - 6*x*Sinh[x] + (3*x*Cosh[x]*Sinh[x])/2 - (3*Sinh
[x]^2)/4

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2633

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]

Rubi steps

\begin{align*} \int (x+\sinh (x))^3 \, dx &=\int \left (x^3+3 x^2 \sinh (x)+3 x \sinh ^2(x)+\sinh ^3(x)\right ) \, dx\\ &=\frac{x^4}{4}+3 \int x^2 \sinh (x) \, dx+3 \int x \sinh ^2(x) \, dx+\int \sinh ^3(x) \, dx\\ &=\frac{x^4}{4}+3 x^2 \cosh (x)+\frac{3}{2} x \cosh (x) \sinh (x)-\frac{3 \sinh ^2(x)}{4}-\frac{3 \int x \, dx}{2}-6 \int x \cosh (x) \, dx-\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=-\frac{3 x^2}{4}+\frac{x^4}{4}-\cosh (x)+3 x^2 \cosh (x)+\frac{\cosh ^3(x)}{3}-6 x \sinh (x)+\frac{3}{2} x \cosh (x) \sinh (x)-\frac{3 \sinh ^2(x)}{4}+6 \int \sinh (x) \, dx\\ &=-\frac{3 x^2}{4}+\frac{x^4}{4}+5 \cosh (x)+3 x^2 \cosh (x)+\frac{\cosh ^3(x)}{3}-6 x \sinh (x)+\frac{3}{2} x \cosh (x) \sinh (x)-\frac{3 \sinh ^2(x)}{4}\\ \end{align*}

Mathematica [A]  time = 0.084436, size = 48, normalized size = 0.86 \[ \frac{1}{24} \left (6 x \left (x^3-3 x-24 \sinh (x)+3 \sinh (2 x)\right )+18 \left (4 x^2+7\right ) \cosh (x)-9 \cosh (2 x)+2 \cosh (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x + Sinh[x])^3,x]

[Out]

(18*(7 + 4*x^2)*Cosh[x] - 9*Cosh[2*x] + 2*Cosh[3*x] + 6*x*(-3*x + x^3 - 24*Sinh[x] + 3*Sinh[2*x]))/24

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Maple [A]  time = 0.008, size = 52, normalized size = 0.9 \begin{align*} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( x \right ) +{\frac{3\,x\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}-{\frac{3\,{x}^{2}}{4}}-{\frac{3\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{4}}+3\,{x}^{2}\cosh \left ( x \right ) -6\,x\sinh \left ( x \right ) +6\,\cosh \left ( x \right ) +{\frac{{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+sinh(x))^3,x)

[Out]

(-2/3+1/3*sinh(x)^2)*cosh(x)+3/2*x*cosh(x)*sinh(x)-3/4*x^2-3/4*cosh(x)^2+3*x^2*cosh(x)-6*x*sinh(x)+6*cosh(x)+1
/4*x^4

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Maxima [A]  time = 1.03458, size = 109, normalized size = 1.95 \begin{align*} \frac{1}{4} \, x^{4} - \frac{3}{4} \, x^{2} + \frac{3}{16} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac{3}{2} \,{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac{3}{16} \,{\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac{3}{2} \,{\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac{1}{24} \, e^{\left (3 \, x\right )} - \frac{3}{8} \, e^{\left (-x\right )} + \frac{1}{24} \, e^{\left (-3 \, x\right )} - \frac{3}{8} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+sinh(x))^3,x, algorithm="maxima")

[Out]

1/4*x^4 - 3/4*x^2 + 3/16*(2*x - 1)*e^(2*x) + 3/2*(x^2 + 2*x + 2)*e^(-x) - 3/16*(2*x + 1)*e^(-2*x) + 3/2*(x^2 -
2*x + 2)*e^x + 1/24*e^(3*x) - 3/8*e^(-x) + 1/24*e^(-3*x) - 3/8*e^x

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Fricas [A]  time = 2.06957, size = 192, normalized size = 3.43 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{12} \, \cosh \left (x\right )^{3} + \frac{1}{8} \,{\left (2 \, \cosh \left (x\right ) - 3\right )} \sinh \left (x\right )^{2} - \frac{3}{4} \, x^{2} + \frac{3}{4} \,{\left (4 \, x^{2} + 7\right )} \cosh \left (x\right ) - \frac{3}{8} \, \cosh \left (x\right )^{2} + \frac{3}{2} \,{\left (x \cosh \left (x\right ) - 4 \, x\right )} \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+sinh(x))^3,x, algorithm="fricas")

[Out]

1/4*x^4 + 1/12*cosh(x)^3 + 1/8*(2*cosh(x) - 3)*sinh(x)^2 - 3/4*x^2 + 3/4*(4*x^2 + 7)*cosh(x) - 3/8*cosh(x)^2 +
3/2*(x*cosh(x) - 4*x)*sinh(x)

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Sympy [A]  time = 0.455458, size = 85, normalized size = 1.52 \begin{align*} \frac{x^{4}}{4} + \frac{3 x^{2} \sinh ^{2}{\left (x \right )}}{4} - \frac{3 x^{2} \cosh ^{2}{\left (x \right )}}{4} + 3 x^{2} \cosh{\left (x \right )} + \frac{3 x \sinh{\left (x \right )} \cosh{\left (x \right )}}{2} - 6 x \sinh{\left (x \right )} + \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )} - \frac{2 \cosh ^{3}{\left (x \right )}}{3} - \frac{3 \cosh ^{2}{\left (x \right )}}{4} + 6 \cosh{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+sinh(x))**3,x)

[Out]

x**4/4 + 3*x**2*sinh(x)**2/4 - 3*x**2*cosh(x)**2/4 + 3*x**2*cosh(x) + 3*x*sinh(x)*cosh(x)/2 - 6*x*sinh(x) + si
nh(x)**2*cosh(x) - 2*cosh(x)**3/3 - 3*cosh(x)**2/4 + 6*cosh(x)

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Giac [A]  time = 1.24664, size = 101, normalized size = 1.8 \begin{align*} \frac{1}{4} \, x^{4} - \frac{3}{4} \, x^{2} + \frac{3}{16} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac{3}{8} \,{\left (4 \, x^{2} + 8 \, x + 7\right )} e^{\left (-x\right )} - \frac{3}{16} \,{\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac{3}{8} \,{\left (4 \, x^{2} - 8 \, x + 7\right )} e^{x} + \frac{1}{24} \, e^{\left (3 \, x\right )} + \frac{1}{24} \, e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+sinh(x))^3,x, algorithm="giac")

[Out]

1/4*x^4 - 3/4*x^2 + 3/16*(2*x - 1)*e^(2*x) + 3/8*(4*x^2 + 8*x + 7)*e^(-x) - 3/16*(2*x + 1)*e^(-2*x) + 3/8*(4*x
^2 - 8*x + 7)*e^x + 1/24*e^(3*x) + 1/24*e^(-3*x)