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

Optimal. Leaf size=22 \[ \sqrt{\sin ^2(x)}-x \sqrt{\sin ^2(x)} \cot (x) \]

[Out]

Sqrt[Sin[x]^2] - x*Cot[x]*Sqrt[Sin[x]^2]

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Rubi [A]  time = 0.0545788, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6720, 3296, 2637} \[ \sqrt{\sin ^2(x)}-x \sqrt{\sin ^2(x)} \cot (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[Sin[x]^2] - x*Cot[x]*Sqrt[Sin[x]^2]

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0161902, size = 16, normalized size = 0.73 \[ \sqrt{\sin ^2(x)} (1-x \cot (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - x*Cot[x])*Sqrt[Sin[x]^2]

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Maple [C]  time = 0.147, size = 75, normalized size = 3.4 \begin{align*}{\frac{-{\frac{i}{2}}{{\rm e}^{2\,ix}} \left ( x+i \right ) }{{{\rm e}^{2\,ix}}-1}\sqrt{- \left ({{\rm e}^{2\,ix}}-1 \right ) ^{2}{{\rm e}^{-2\,ix}}}}-{\frac{{\frac{i}{2}} \left ( x-i \right ) }{{{\rm e}^{2\,ix}}-1}\sqrt{- \left ({{\rm e}^{2\,ix}}-1 \right ) ^{2}{{\rm e}^{-2\,ix}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*(-(exp(2*I*x)-1)^2*exp(-2*I*x))^(1/2)/(exp(2*I*x)-1)*exp(2*I*x)*(x+I)-1/2*I*(-(exp(2*I*x)-1)^2*exp(-2*I
*x))^(1/2)/(exp(2*I*x)-1)*(x-I)

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Maxima [A]  time = 1.70675, size = 12, normalized size = 0.55 \begin{align*} x \cos \left (x\right ) - \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*cos(x) - sin(x)

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Fricas [A]  time = 0.461237, size = 27, normalized size = 1.23 \begin{align*} -x \cos \left (x\right ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*cos(x) + sin(x)

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Sympy [A]  time = 2.08574, size = 24, normalized size = 1.09 \begin{align*} - \frac{x \sqrt{\sin ^{2}{\left (x \right )}} \cos{\left (x \right )}}{\sin{\left (x \right )}} + \sqrt{\sin ^{2}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*sqrt(sin(x)**2)*cos(x)/sin(x) + sqrt(sin(x)**2)

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Giac [A]  time = 1.11372, size = 20, normalized size = 0.91 \begin{align*} -x \cos \left (x\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) + \mathrm{sgn}\left (\sin \left (x\right )\right ) \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*cos(x)*sgn(sin(x)) + sgn(sin(x))*sin(x)