3.12 \(\int \frac{\sin (\sqrt{1+x})}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=10 \[ -2 \cos \left (\sqrt{x+1}\right ) \]

[Out]

-2*Cos[Sqrt[1 + x]]

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Rubi [A]  time = 0.0162841, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3431, 15, 2638} \[ -2 \cos \left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[Sqrt[1 + x]]/Sqrt[1 + x],x]

[Out]

-2*Cos[Sqrt[1 + x]]

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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 \frac{\sin \left (\sqrt{1+x}\right )}{\sqrt{1+x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \sin (x)}{\sqrt{x^2}} \, dx,x,\sqrt{1+x}\right )\\ &=2 \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sqrt{1+x}\right )\\ &=-2 \cos \left (\sqrt{1+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0184512, size = 10, normalized size = 1. \[ -2 \cos \left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[Sqrt[1 + x]]/Sqrt[1 + x],x]

[Out]

-2*Cos[Sqrt[1 + x]]

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Maple [A]  time = 0.007, size = 9, normalized size = 0.9 \begin{align*} -2\,\cos \left ( \sqrt{1+x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin((1+x)^(1/2))/(1+x)^(1/2),x)

[Out]

-2*cos((1+x)^(1/2))

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Maxima [A]  time = 0.997247, size = 11, normalized size = 1.1 \begin{align*} -2 \, \cos \left (\sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin((1+x)^(1/2))/(1+x)^(1/2),x, algorithm="maxima")

[Out]

-2*cos(sqrt(x + 1))

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Fricas [A]  time = 0.453937, size = 28, normalized size = 2.8 \begin{align*} -2 \, \cos \left (\sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin((1+x)^(1/2))/(1+x)^(1/2),x, algorithm="fricas")

[Out]

-2*cos(sqrt(x + 1))

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Sympy [A]  time = 0.31801, size = 10, normalized size = 1. \begin{align*} - 2 \cos{\left (\sqrt{x + 1} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin((1+x)**(1/2))/(1+x)**(1/2),x)

[Out]

-2*cos(sqrt(x + 1))

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Giac [A]  time = 1.08254, size = 11, normalized size = 1.1 \begin{align*} -2 \, \cos \left (\sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin((1+x)^(1/2))/(1+x)^(1/2),x, algorithm="giac")

[Out]

-2*cos(sqrt(x + 1))