\(\int \frac {\sin (\sqrt {1+x})}{\sqrt {1+x}} \, dx\) [12]

Optimal result

Integrand size = 16, antiderivative size = 10 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \cos \left (\sqrt {1+x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3512, 15, 2718} \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \cos \left (\sqrt {x+1}\right ) \]

[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 2718

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

Rule 3512

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \cos \left (\sqrt {1+x}\right ) \]

[In]

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

[Out]

-2*Cos[Sqrt[1 + x]]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(-2 \cos \left (\sqrt {1+x}\right )\)	\(9\)
default	\(-2 \cos \left (\sqrt {1+x}\right )\)	\(9\)

[In]

int(sin((1+x)^(1/2))/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \, \cos \left (\sqrt {x + 1}\right ) \]

[In]

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

[Out]

-2*cos(sqrt(x + 1))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=- 2 \cos {\left (\sqrt {x + 1} \right )} \]

[In]

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

[Out]

-2*cos(sqrt(x + 1))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \, \cos \left (\sqrt {x + 1}\right ) \]

[In]

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

[Out]

-2*cos(sqrt(x + 1))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \, \cos \left (\sqrt {x + 1}\right ) \]

[In]

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

[Out]

-2*cos(sqrt(x + 1))

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2\,\cos \left (\sqrt {x+1}\right ) \]

[In]

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

[Out]

-2*cos((x + 1)^(1/2))