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

Optimal. Leaf size=45 \[ -\frac{1}{4} \sin ^{-1}(\cos (x)-\sin (x))-\frac{1}{2} \sqrt{\sin (2 x)} \cos (x)+\frac{1}{4} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right ) \]

[Out]

-ArcSin[Cos[x] - Sin[x]]/4 + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/4 - (Cos[x]*Sqrt[Sin[2*x]])/2

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Rubi [A]  time = 0.0297949, antiderivative size = 45, 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 = {4302, 4305} \[ -\frac{1}{4} \sin ^{-1}(\cos (x)-\sin (x))-\frac{1}{2} \sqrt{\sin (2 x)} \cos (x)+\frac{1}{4} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcSin[Cos[x] - Sin[x]]/4 + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/4 - (Cos[x]*Sqrt[Sin[2*x]])/2

Rule 4302

Int[sin[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(-2*Cos[a + b*x]*(g*Sin[c
+ d*x])^p)/(d*(2*p + 1)), x] + Dist[(2*p*g)/(2*p + 1), Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; Fr
eeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p] && GtQ[p, 0] && 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 \sin (x) \sqrt{\sin (2 x)} \, dx &=-\frac{1}{2} \cos (x) \sqrt{\sin (2 x)}+\frac{1}{2} \int \frac{\cos (x)}{\sqrt{\sin (2 x)}} \, dx\\ &=-\frac{1}{4} \sin ^{-1}(\cos (x)-\sin (x))+\frac{1}{4} \log \left (\cos (x)+\sin (x)+\sqrt{\sin (2 x)}\right )-\frac{1}{2} \cos (x) \sqrt{\sin (2 x)}\\ \end{align*}

Mathematica [A]  time = 0.0352929, size = 41, normalized size = 0.91 \[ \frac{1}{4} \left (-\sin ^{-1}(\cos (x)-\sin (x))-2 \sqrt{\sin (2 x)} \cos (x)+\log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]] - 2*Cos[x]*Sqrt[Sin[2*x]])/4

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Maple [C]  time = 0.053, size = 171, normalized size = 3.8 \begin{align*}{\sqrt{-{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ) \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ) +2\, \left ( \tan \left ( x/2 \right ) \right ) ^{3}-2\,\tan \left ( x/2 \right ) \right ){\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{x}{2}} \right ) }}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(2*x))*sin(x), x)

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Fricas [B]  time = 1.97212, size = 513, normalized size = 11.4 \begin{align*} -\frac{1}{2} \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} \cos \left (x\right ) + \frac{1}{8} \, \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}{8} \, \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}{16} \, \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(sin(x)*sin(2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*sqrt(cos(x)*sin(x))*cos(x) + 1/8*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)) - 1/8*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos
(x) - sin(x))) - 1/16*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(sin(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 \sqrt{\sin \left (2 \, x\right )} \sin \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(2*x))*sin(x), x)