3.32 \(\int (a-a \sin ^2(x))^2 \, dx\)

Optimal. Leaf size=33 \[ \frac{3 a^2 x}{8}+\frac{1}{4} a^2 \sin (x) \cos ^3(x)+\frac{3}{8} a^2 \sin (x) \cos (x) \]

[Out]

(3*a^2*x)/8 + (3*a^2*Cos[x]*Sin[x])/8 + (a^2*Cos[x]^3*Sin[x])/4

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Rubi [A]  time = 0.0256267, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3175, 2635, 8} \[ \frac{3 a^2 x}{8}+\frac{1}{4} a^2 \sin (x) \cos ^3(x)+\frac{3}{8} a^2 \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sin[x]^2)^2,x]

[Out]

(3*a^2*x)/8 + (3*a^2*Cos[x]*Sin[x])/8 + (a^2*Cos[x]^3*Sin[x])/4

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \left (a-a \sin ^2(x)\right )^2 \, dx &=a^2 \int \cos ^4(x) \, dx\\ &=\frac{1}{4} a^2 \cos ^3(x) \sin (x)+\frac{1}{4} \left (3 a^2\right ) \int \cos ^2(x) \, dx\\ &=\frac{3}{8} a^2 \cos (x) \sin (x)+\frac{1}{4} a^2 \cos ^3(x) \sin (x)+\frac{1}{8} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{3 a^2 x}{8}+\frac{3}{8} a^2 \cos (x) \sin (x)+\frac{1}{4} a^2 \cos ^3(x) \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0027889, size = 26, normalized size = 0.79 \[ a^2 \left (\frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sin[x]^2)^2,x]

[Out]

a^2*((3*x)/8 + Sin[2*x]/4 + Sin[4*x]/32)

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Maple [A]  time = 0.021, size = 43, normalized size = 1.3 \begin{align*}{a}^{2} \left ( -{\frac{\cos \left ( x \right ) }{4} \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{3\,\sin \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{8}} \right ) -2\,{a}^{2} \left ( -1/2\,\sin \left ( x \right ) \cos \left ( x \right ) +x/2 \right ) +{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*sin(x)^2)^2,x)

[Out]

a^2*(-1/4*(sin(x)^3+3/2*sin(x))*cos(x)+3/8*x)-2*a^2*(-1/2*sin(x)*cos(x)+1/2*x)+a^2*x

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Maxima [A]  time = 0.948838, size = 54, normalized size = 1.64 \begin{align*} \frac{1}{32} \, a^{2}{\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - \frac{1}{2} \, a^{2}{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

1/32*a^2*(12*x + sin(4*x) - 8*sin(2*x)) - 1/2*a^2*(2*x - sin(2*x)) + a^2*x

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Fricas [A]  time = 1.64587, size = 76, normalized size = 2.3 \begin{align*} \frac{3}{8} \, a^{2} x + \frac{1}{8} \,{\left (2 \, a^{2} \cos \left (x\right )^{3} + 3 \, a^{2} \cos \left (x\right )\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

3/8*a^2*x + 1/8*(2*a^2*cos(x)^3 + 3*a^2*cos(x))*sin(x)

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Sympy [B]  time = 1.06638, size = 110, normalized size = 3.33 \begin{align*} \frac{3 a^{2} x \sin ^{4}{\left (x \right )}}{8} + \frac{3 a^{2} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} - a^{2} x \sin ^{2}{\left (x \right )} + \frac{3 a^{2} x \cos ^{4}{\left (x \right )}}{8} - a^{2} x \cos ^{2}{\left (x \right )} + a^{2} x - \frac{5 a^{2} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{8} - \frac{3 a^{2} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + a^{2} \sin{\left (x \right )} \cos{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(x)**2)**2,x)

[Out]

3*a**2*x*sin(x)**4/8 + 3*a**2*x*sin(x)**2*cos(x)**2/4 - a**2*x*sin(x)**2 + 3*a**2*x*cos(x)**4/8 - a**2*x*cos(x
)**2 + a**2*x - 5*a**2*sin(x)**3*cos(x)/8 - 3*a**2*sin(x)*cos(x)**3/8 + a**2*sin(x)*cos(x)

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Giac [A]  time = 1.12513, size = 34, normalized size = 1.03 \begin{align*} \frac{3}{8} \, a^{2} x + \frac{1}{32} \, a^{2} \sin \left (4 \, x\right ) + \frac{1}{4} \, a^{2} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

3/8*a^2*x + 1/32*a^2*sin(4*x) + 1/4*a^2*sin(2*x)