3.28 \(\int x^3 \sinh (a+b x^4) \, dx\)

Optimal. Leaf size=15 \[ \frac{\cosh \left (a+b x^4\right )}{4 b} \]

[Out]

Cosh[a + b*x^4]/(4*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0203076, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5320, 2638} \[ \frac{\cosh \left (a+b x^4\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sinh[a + b*x^4],x]

[Out]

Cosh[a + b*x^4]/(4*b)

Rule 5320

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0082737, size = 15, normalized size = 1. \[ \frac{\cosh \left (a+b x^4\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sinh[a + b*x^4],x]

[Out]

Cosh[a + b*x^4]/(4*b)

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \begin{align*}{\frac{\cosh \left ( b{x}^{4}+a \right ) }{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sinh(b*x^4+a),x)

[Out]

1/4*cosh(b*x^4+a)/b

________________________________________________________________________________________

Maxima [A]  time = 1.03286, size = 18, normalized size = 1.2 \begin{align*} \frac{\cosh \left (b x^{4} + a\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sinh(b*x^4+a),x, algorithm="maxima")

[Out]

1/4*cosh(b*x^4 + a)/b

________________________________________________________________________________________

Fricas [A]  time = 1.7345, size = 31, normalized size = 2.07 \begin{align*} \frac{\cosh \left (b x^{4} + a\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sinh(b*x^4+a),x, algorithm="fricas")

[Out]

1/4*cosh(b*x^4 + a)/b

________________________________________________________________________________________

Sympy [A]  time = 1.14299, size = 19, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\cosh{\left (a + b x^{4} \right )}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*sinh(b*x**4+a),x)

[Out]

Piecewise((cosh(a + b*x**4)/(4*b), Ne(b, 0)), (x**4*sinh(a)/4, True))

________________________________________________________________________________________

Giac [A]  time = 1.23275, size = 34, normalized size = 2.27 \begin{align*} \frac{e^{\left (b x^{4} + a\right )} + e^{\left (-b x^{4} - a\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sinh(b*x^4+a),x, algorithm="giac")

[Out]

1/8*(e^(b*x^4 + a) + e^(-b*x^4 - a))/b