3.860 \(\int \frac{\cos (x)+\sin (x)}{\sqrt{1+\sin (2 x)}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x \sqrt{\sin (2 x)+1}}{\sin (x)+\cos (x)} \]

[Out]

(x*Sqrt[1 + Sin[2*x]])/(Cos[x] + Sin[x])

________________________________________________________________________________________

Rubi [B]  time = 1.70716, antiderivative size = 72, normalized size of antiderivative = 3.79, number of steps used = 17, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {4401, 6719, 1075, 628, 635, 203, 260, 12, 1023} \[ \frac{2 \cos ^2\left (\frac{x}{2}\right ) \tan ^{-1}\left (\tan \left (\frac{x}{2}\right )\right ) \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[Tan[x/2]]*Cos[x/2]^2*(1 + 2*Tan[x/2] - Tan[x/2]^2))/Sqrt[Cos[x/2]^4*(1 + 2*Tan[x/2] - Tan[x/2]^2)^2]

Rule 4401

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 1075

Int[((A_.) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> With[{q =
c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*A*c*f + a^2*C*f + c*(
-(b*C*d) + A*b*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - A*c*d*f - a*C*d*f + a*A*f^2 - f*(-(b
*C*d) + A*b*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1023

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> With[{q = Si
mplify[c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2]}, Dist[1/q, Int[Simp[g*c^2*d + g*b^2*f - a*b*h*f - a*g*c*f + c
*(h*c*d + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[Simp[b*h*d*f - g*c*d*f + a*g*f^2 - f*
(h*c*d + g*b*f - a*h*f)*x, x]/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2
- 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{\cos (x)+\sin (x)}{\sqrt{1+\sin (2 x)}} \, dx &=\int \left (\frac{\cos (x)}{\sqrt{1+\sin (2 x)}}+\frac{\sin (x)}{\sqrt{1+\sin (2 x)}}\right ) \, dx\\ &=\int \frac{\cos (x)}{\sqrt{1+\sin (2 x)}} \, dx+\int \frac{\sin (x)}{\sqrt{1+\sin (2 x)}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{2 x}{\left (1+x^2\right )^2 \sqrt{\frac{\left (-1-2 x+x^2\right )^2}{\left (1+x^2\right )^2}}} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+2 \operatorname{Subst}\left (\int \frac{1-x^2}{\left (1+x^2\right )^2 \sqrt{\frac{\left (-1-2 x+x^2\right )^2}{\left (1+x^2\right )^2}}} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^2 \sqrt{\frac{\left (-1-2 x+x^2\right )^2}{\left (1+x^2\right )^2}}} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{\left (2 \cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{\left (1+x^2\right ) \left (-1-2 x+x^2\right )} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}\\ &=\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{-4+4 x}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{4 \sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}+\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{4-4 x}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{4 \sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}+\frac{\left (4 \cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \left (-1-2 x+x^2\right )} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}\\ &=\frac{\cos ^2\left (\frac{x}{2}\right ) \log \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}{2 \sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}}+\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{-2-2 x}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{2 \sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}+\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{-2+2 x}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{2 \sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}-\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}+\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}\\ &=\frac{x \cos ^2\left (\frac{x}{2}\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}{2 \sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}}+\frac{\cos ^2\left (\frac{x}{2}\right ) \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}}-\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}-\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (-1-2 \tan \left (\frac{x}{2}\right )+\tan ^2\left (\frac{x}{2}\right )\right )^2}}\\ &=\frac{x \cos ^2\left (\frac{x}{2}\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^4\left (\frac{x}{2}\right ) \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}}\\ \end{align*}

Mathematica [A]  time = 0.0131581, size = 17, normalized size = 0.89 \[ \frac{x (\sin (x)+\cos (x))}{\sqrt{\sin (2 x)+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(Cos[x] + Sin[x]))/Sqrt[1 + Sin[2*x]]

________________________________________________________________________________________

Maple [C]  time = 0.287, size = 12372, normalized size = 651.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 1.66231, size = 444, normalized size = 23.37 \begin{align*} \frac{1}{16} \, \sqrt{2}{\left (2 \, \sqrt{2} \arctan \left (\sin \left (2 \, x\right ) + 1, \cos \left (2 \, x\right )\right ) + \sqrt{2} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) + 1\right ) + 4 \,{\left (\cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) \sin \left (4 \, x\right ) + \sin \left (4 \, x\right )^{2} - 4 \, \cos \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2}\right )}^{\frac{1}{4}}{\left (\cos \left (\frac{1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right ) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \left (\frac{1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right )\right )}\right )} + \frac{1}{16} \, \sqrt{2}{\left (2 \, \sqrt{2} \arctan \left (\sin \left (2 \, x\right ) + 1, \cos \left (2 \, x\right )\right ) - \sqrt{2} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) + 1\right ) - 4 \,{\left (\cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) \sin \left (4 \, x\right ) + \sin \left (4 \, x\right )^{2} - 4 \, \cos \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2}\right )}^{\frac{1}{4}}{\left (\cos \left (2 \, x\right ) \cos \left (\frac{1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right ) - \sin \left (2 \, x\right ) \sin \left (\frac{1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right )\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*(2*sqrt(2)*arctan2(sin(2*x) + 1, cos(2*x)) + sqrt(2)*log(cos(2*x)^2 + sin(2*x)^2 + 2*sin(2*x) + 1
) + 4*(cos(4*x)^2 + 4*cos(2*x)^2 + 4*cos(2*x)*sin(4*x) + sin(4*x)^2 - 4*cos(4*x)*sin(2*x) + 4*sin(2*x)^2)^(1/4
)*(cos(1/2*arctan2(cos(4*x) - 2*sin(2*x), 2*cos(2*x) + sin(4*x)))*sin(2*x) + cos(2*x)*sin(1/2*arctan2(cos(4*x)
- 2*sin(2*x), 2*cos(2*x) + sin(4*x))))) + 1/16*sqrt(2)*(2*sqrt(2)*arctan2(sin(2*x) + 1, cos(2*x)) - sqrt(2)*l
og(cos(2*x)^2 + sin(2*x)^2 + 2*sin(2*x) + 1) - 4*(cos(4*x)^2 + 4*cos(2*x)^2 + 4*cos(2*x)*sin(4*x) + sin(4*x)^2
- 4*cos(4*x)*sin(2*x) + 4*sin(2*x)^2)^(1/4)*(cos(2*x)*cos(1/2*arctan2(cos(4*x) - 2*sin(2*x), 2*cos(2*x) + sin
(4*x))) - sin(2*x)*sin(1/2*arctan2(cos(4*x) - 2*sin(2*x), 2*cos(2*x) + sin(4*x)))))

________________________________________________________________________________________

Fricas [A]  time = 1.96431, size = 5, normalized size = 0.26 \begin{align*} -x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (x \right )} + \cos{\left (x \right )}}{\sqrt{\sin{\left (2 x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((sin(x) + cos(x))/sqrt(sin(2*x) + 1), x)

________________________________________________________________________________________

Giac [B]  time = 1.11637, size = 57, normalized size = 3. \begin{align*} \frac{2 \, \pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor - x}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, x\right )^{3} - 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*pi*floor(1/2*x/pi + 1/2) - x)/sgn(tan(1/2*x)^4 - 2*tan(1/2*x)^3 - 2*tan(1/2*x) - 1)