3.13 \(\int x^{-1+n} \sin (x^n) \, dx\)

Optimal. Leaf size=9 \[ -\frac{\cos \left (x^n\right )}{n} \]

[Out]

-(Cos[x^n]/n)

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Rubi [A]  time = 0.0111279, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3379, 2638} \[ -\frac{\cos \left (x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*Sin[x^n],x]

[Out]

-(Cos[x^n]/n)

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(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^{-1+n} \sin \left (x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sin (x) \, dx,x,x^n\right )}{n}\\ &=-\frac{\cos \left (x^n\right )}{n}\\ \end{align*}

Mathematica [A]  time = 0.0137676, size = 9, normalized size = 1. \[ -\frac{\cos \left (x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*Sin[x^n],x]

[Out]

-(Cos[x^n]/n)

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Maple [A]  time = 0.005, size = 10, normalized size = 1.1 \begin{align*} -{\frac{\cos \left ({x}^{n} \right ) }{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*sin(x^n),x)

[Out]

-cos(x^n)/n

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Maxima [A]  time = 0.94586, size = 12, normalized size = 1.33 \begin{align*} -\frac{\cos \left (x^{n}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*sin(x^n),x, algorithm="maxima")

[Out]

-cos(x^n)/n

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Fricas [A]  time = 0.46481, size = 18, normalized size = 2. \begin{align*} -\frac{\cos \left (x^{n}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*sin(x^n),x, algorithm="fricas")

[Out]

-cos(x^n)/n

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Sympy [A]  time = 17.818, size = 7, normalized size = 0.78 \begin{align*} - \frac{\cos{\left (x^{n} \right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*sin(x**n),x)

[Out]

-cos(x**n)/n

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Giac [A]  time = 1.07018, size = 12, normalized size = 1.33 \begin{align*} -\frac{\cos \left (x^{n}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*sin(x^n),x, algorithm="giac")

[Out]

-cos(x^n)/n