3.186 \(\int \csc (x) \sqrt{\sin (2 x)} \, dx\)

Optimal. Leaf size=25 \[ \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )-\sin ^{-1}(\cos (x)-\sin (x)) \]

[Out]

-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]

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Rubi [A]  time = 0.0303279, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4308, 4305} \[ \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )-\sin ^{-1}(\cos (x)-\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]*Sqrt[Sin[2*x]],x]

[Out]

-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]

Rule 4308

Int[((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_)/sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Dist[2*g, Int[Cos[a + b*x]*(g*S
in[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ
[p] && IntegerQ[2*p]

Rule 4305

Int[cos[(a_.) + (b_.)*(x_)]/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> -Simp[ArcSin[Cos[a + b*x] - Sin[a + b*
x]]/d, x] + Simp[Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[
b*c - a*d, 0] && EqQ[d/b, 2]

Rubi steps

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

Mathematica [A]  time = 0.0164929, size = 25, normalized size = 1. \[ \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )-\sin ^{-1}(\cos (x)-\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]*Sqrt[Sin[2*x]],x]

[Out]

-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]

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Maple [C]  time = 0.046, size = 99, normalized size = 4. \begin{align*} 2\,{\frac{ \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}-1 \right ) \sqrt{\tan \left ( x/2 \right ) +1}\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ( x/2 \right ) }{\it EllipticF} \left ( \sqrt{\tan \left ( x/2 \right ) +1},1/2\,\sqrt{2} \right ) }{\sqrt{\tan \left ( x/2 \right ) \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}-1 \right ) }\sqrt{ \left ( \tan \left ( x/2 \right ) \right ) ^{3}-\tan \left ( x/2 \right ) }}\sqrt{-{\frac{\tan \left ( x/2 \right ) }{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)*sin(2*x)^(1/2),x)

[Out]

2*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)/(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1)^(1/
2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)*EllipticF((tan(1/2*x)+1)^(1/2),
1/2*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (x\right ) \sqrt{\sin \left (2 \, x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(x)*sqrt(sin(2*x)), x)

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Fricas [B]  time = 0.508127, size = 455, normalized size = 18.2 \begin{align*} \frac{1}{2} \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac{1}{2} \, \arctan \left (-\frac{2 \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac{1}{4} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt{2}{\left (4 \, \cos \left (x\right )^{3} -{\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt{\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (x\right ) \sqrt{\sin \left (2 \, x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(x)*sqrt(sin(2*x)), x)