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((1+x)^(1/2))

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Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 10, normalized size = 1.00 \[ -2 \cos \left (\sqrt {x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[Sqrt[1 + x]]

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fricas [A]  time = 0.41, size = 8, normalized size = 0.80 \[ -2 \, \cos \left (\sqrt {x + 1}\right ) \]

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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giac [A]  time = 0.01, size = 8, normalized size = 0.80 \[ -2 \, \cos \left (\sqrt {x + 1}\right ) \]

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))

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maple [A]  time = 0.01, size = 9, normalized size = 0.90 \[ -2 \cos \left (\sqrt {x +1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 8, normalized size = 0.80 \[ -2 \, \cos \left (\sqrt {x + 1}\right ) \]

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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mupad [B]  time = 0.17, size = 8, normalized size = 0.80 \[ -2\,\cos \left (\sqrt {x+1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos((x + 1)^(1/2))

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sympy [A]  time = 0.33, size = 10, normalized size = 1.00 \[ - 2 \cos {\left (\sqrt {x + 1} \right )} \]

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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