3.48 \(\int \frac{x \cos (\sqrt{1+x^2})}{\sqrt{1+x^2}} \, dx\)

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

[Out]

Sin[Sqrt[1 + x^2]]

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Rubi [A]  time = 0.135869, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6715, 3432, 15, 2637} \[ \sin \left (\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x*Cos[Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]

[Out]

Sin[Sqrt[1 + x^2]]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 3432

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Cos[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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0324673, size = 10, normalized size = 1. \[ \sin \left (\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Cos[Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]

[Out]

Sin[Sqrt[1 + x^2]]

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Maple [A]  time = 0.011, size = 9, normalized size = 0.9 \begin{align*} \sin \left ( \sqrt{{x}^{2}+1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos((x^2+1)^(1/2))/(x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.12438, size = 11, normalized size = 1.1 \begin{align*} \sin \left (\sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sqrt(x^2 + 1))

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Fricas [A]  time = 2.30427, size = 27, normalized size = 2.7 \begin{align*} \sin \left (\sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sqrt(x^2 + 1))

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Sympy [A]  time = 0.902063, size = 8, normalized size = 0.8 \begin{align*} \sin{\left (\sqrt{x^{2} + 1} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sqrt(x**2 + 1))

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Giac [A]  time = 1.09923, size = 11, normalized size = 1.1 \begin{align*} \sin \left (\sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sqrt(x^2 + 1))