3.3.89 \(\int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx\) [289]

Optimal. Leaf size=122 \[ -\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{2 \sqrt {2}} \]

[Out]

-1/2*arctan(1-2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/2)+1/2*arctan(1+2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/
2)+1/4*ln(1-2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2)+tan(x))*2^(1/2)-1/4*ln(1+2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2)+tan(x
))*2^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2654, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\text {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sin[x]]/Sqrt[Cos[x]],x]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/S
qrt[2] + Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]) - Log[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sq
rt[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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.01, size = 38, normalized size = 0.31 \begin {gather*} \frac {2 \cos ^2(x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\sin ^2(x)\right ) \sin ^{\frac {3}{2}}(x)}{3 \cos ^{\frac {3}{2}}(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sin[x]]/Sqrt[Cos[x]],x]

[Out]

(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[x]^2]*Sin[x]^(3/2))/(3*Cos[x]^(3/2))

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.11, size = 166, normalized size = 1.36

method	result	size
default	\(-\frac {\left (\sin ^{\frac {3}{2}}\left (x \right )\right ) \left (i \EllipticPi \left (\sqrt {\frac {1-\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \EllipticPi \left (\sqrt {\frac {1-\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-\EllipticPi \left (\sqrt {\frac {1-\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-\EllipticPi \left (\sqrt {\frac {1-\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\frac {-1+\cos \left (x \right )}{\sin \left (x \right )}}\, \sqrt {\frac {-1+\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}\, \sqrt {\frac {1-\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}\, \sqrt {2}}{2 \left (-1+\cos \left (x \right )\right ) \sqrt {\cos \left (x \right )}}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sin(x)^(3/2)*(I*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-I*EllipticPi(((1-cos(x
)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/
2))-EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2)))*((-1+cos(x))/sin(x))^(1/2)*((-1+cos(x)
+sin(x))/sin(x))^(1/2)*((1-cos(x)+sin(x))/sin(x))^(1/2)/(-1+cos(x))/cos(x)^(1/2)*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (87) = 174\).
time = 0.67, size = 447, normalized size = 3.66 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {2} \sqrt {2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - 2 \, \cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {2 \, \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} \sin \left (x\right ) - \sqrt {2} \sqrt {-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - 2 \, \cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} {\left (\sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + \cos \left (x\right ) + \sin \left (x\right )\right )} + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} {\left (\sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )\right )} + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*arctan(1/2*(2*cos(x)^3 - 2*cos(x)^2*sin(x) + sqrt(2)*sqrt(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt
(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)*sqrt(cos(x))*sqrt(sin(x)) - sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) - 2
*cos(x))/(cos(x)^3 + cos(x)^2*sin(x) - cos(x))) + 1/4*sqrt(2)*arctan(-1/2*(2*cos(x)^3 - 2*cos(x)^2*sin(x) - sq
rt(2)*sqrt(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)*sqrt(cos(x))*
sqrt(sin(x)) + sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + cos(x)^2*sin(x) - cos(x))) - 1/4*sqrt
(2)*arctan(-(sqrt(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)*(sqrt(
2)*sqrt(cos(x))*sqrt(sin(x)) + cos(x) + sin(x)) + sqrt(2)*sqrt(cos(x))*sqrt(sin(x)))/(cos(x) - sin(x))) - 1/4*
sqrt(2)*arctan(-(sqrt(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)*(sq
rt(2)*sqrt(cos(x))*sqrt(sin(x)) - cos(x) - sin(x)) + sqrt(2)*sqrt(cos(x))*sqrt(sin(x)))/(cos(x) - sin(x))) - 1
/8*sqrt(2)*log(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1) + 1/8*sqrt
(2)*log(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sin {\left (x \right )}}}{\sqrt {\cos {\left (x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.72, size = 25, normalized size = 0.20 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^(1/2)/cos(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)

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