\(\int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx\) [912]

Optimal result

Integrand size = 18, antiderivative size = 57 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right ) \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(57)=114\).

Time = 0.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.26, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3186, 2655, 303, 1176, 631, 210, 1179, 642, 2654} \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}+\frac {\log \left (\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (x)-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\cot (x)+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

ArcTan[1 - (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/Sqrt[2] - ArcTan[1 + (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/Sqrt
[2] - ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2] + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]
]/Sqrt[2] - Log[1 + Cot[x] - (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/(2*Sqrt[2]) + Log[1 + Cot[x] + (Sqrt[2]*Sqrt
[Cos[x]])/Sqrt[Sin[x]]]/(2*Sqrt[2]) + Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]) - Log[
1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2])

Rule 210

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2654

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[k*a*(b/f), Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos
[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rule 2655

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(-k)*a*(b/f), Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*
Sin[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rule 3186

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c +
d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\frac {2 \sqrt [4]{\cos ^2(x)} \sqrt {\sin (x)} \left (3 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\sin ^2(x)\right )+\sqrt {\cos ^2(x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\sin ^2(x)\right ) \sin (x)\right )}{3 \cos ^{\frac {3}{2}}(x)} \]

[In]

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

[Out]

(2*(Cos[x]^2)^(1/4)*Sqrt[Sin[x]]*(3*Cos[x]*Hypergeometric2F1[1/4, 1/4, 5/4, Sin[x]^2] + Sqrt[Cos[x]^2]*Hyperge
ometric2F1[3/4, 3/4, 7/4, Sin[x]^2]*Sin[x]))/(3*Cos[x]^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(41)=82\).

Time = 24.51 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {\sqrt {2}\, \left (\arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )+\arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )-1}\right )\right ) \left (\cos \left (x \right )-1\right ) \sqrt {\cos \left (x \right )}}{\sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )^{\frac {3}{2}}}\)	\(96\)
parts	\(-\frac {\sqrt {\cos \left (x \right )}\, \left (\cos \left (x \right )-1\right ) \left (\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )+2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )-\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )+2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )-1}\right )\right ) \sqrt {2}}{4 \sin \left (x \right )^{\frac {3}{2}} \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}+\frac {\left (\cos \left (x \right )-1\right ) \sqrt {\cos \left (x \right )}\, \left (\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )-2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )-\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )-2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )-1}\right )\right ) \sqrt {2}}{4 \sin \left (x \right )^{\frac {3}{2}} \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(392\)

[In]

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

[Out]

-2^(1/2)*(arctan((2^(1/2)*sin(x)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)+cos(x)-1)/(cos(x)-1))+arctan((2^(1/2)*sin(
x)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)-1)))*(cos(x)-1)*cos(x)^(1/2)/(cos(x)*sin(x)/(cos(x)+1)
^2)^(1/2)/sin(x)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (41) = 82\).

Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (32 \, \sqrt {2} \cos \left (x\right )^{4} - 32 \, \sqrt {2} \cos \left (x\right )^{2} + 16 \, \sqrt {2} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {2}\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{8 \, {\left (4 \, \cos \left (x\right )^{5} - 3 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{4} - 5 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) \]

[In]

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

[Out]

-1/4*sqrt(2)*arctan(-1/8*(32*sqrt(2)*cos(x)^4 - 32*sqrt(2)*cos(x)^2 + 16*sqrt(2)*cos(x)*sin(x) - sqrt(2))*sqrt
(cos(x))*sqrt(sin(x))/(4*cos(x)^5 - 3*cos(x)^3 - (4*cos(x)^4 - 5*cos(x)^2)*sin(x) - cos(x)))

Sympy [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int \frac {\sin {\left (x \right )} + \cos {\left (x \right )}}{\sqrt {\sin {\left (x \right )}} \sqrt {\cos {\left (x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int { \frac {\cos \left (x\right ) + \sin \left (x\right )}{\sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}} \,d x } \]

[In]

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

[Out]

integrate((cos(x) + sin(x))/(sqrt(cos(x))*sqrt(sin(x))), x)

Giac [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int { \frac {\cos \left (x\right ) + \sin \left (x\right )}{\sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}} \,d x } \]

[In]

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

[Out]

integrate((cos(x) + sin(x))/(sqrt(cos(x))*sqrt(sin(x))), x)

Mupad [B] (verification not implemented)

Time = 28.44 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\frac {2\,\sqrt {\cos \left (x\right )}\,{\sin \left (x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{3/4}}-\frac {2\,{\cos \left (x\right )}^{3/2}\,\sqrt {\sin \left (x\right )}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {3}{4};\ \frac {7}{4};\ {\cos \left (x\right )}^2\right )}{3\,{\left ({\sin \left (x\right )}^2\right )}^{1/4}} \]

[In]

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

[Out]

- (2*cos(x)^(1/2)*sin(x)^(3/2)*hypergeom([1/4, 1/4], 5/4, cos(x)^2))/(sin(x)^2)^(3/4) - (2*cos(x)^(3/2)*sin(x)
^(1/2)*hypergeom([3/4, 3/4], 7/4, cos(x)^2))/(3*(sin(x)^2)^(1/4))