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]

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

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 8

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

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 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]

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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.42, size = 28, normalized size = 0.85 \[ \frac {3}{8} \, a^{2} x + \frac {1}{8} \, {\left (2 \, a^{2} \cos \relax (x)^{3} + 3 \, a^{2} \cos \relax (x)\right )} \sin \relax (x) \]

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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giac [A]  time = 0.14, size = 25, normalized size = 0.76 \[ \frac {3}{8} \, a^{2} x + \frac {1}{32} \, a^{2} \sin \left (4 \, x\right ) + \frac {1}{4} \, a^{2} \sin \left (2 \, x\right ) \]

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)

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maple [A]  time = 0.34, size = 43, normalized size = 1.30 \[ a^{2} \left (-\frac {\left (\sin ^{3}\relax (x )+\frac {3 \sin \relax (x )}{2}\right ) \cos \relax (x )}{4}+\frac {3 x}{8}\right )-2 a^{2} \left (-\frac {\sin \relax (x ) \cos \relax (x )}{2}+\frac {x}{2}\right )+a^{2} x \]

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 = 0.35, size = 40, normalized size = 1.21 \[ \frac {1}{32} \, a^{2} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - \frac {1}{2} \, a^{2} {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{2} x \]

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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mupad [B]  time = 13.76, size = 33, normalized size = 1.00 \[ \frac {\frac {3\,a^2\,{\mathrm {tan}\relax (x)}^3}{8}+\frac {5\,a^2\,\mathrm {tan}\relax (x)}{8}}{{\left ({\mathrm {tan}\relax (x)}^2+1\right )}^2}+\frac {3\,a^2\,x}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*sin(x)^2)^2,x)

[Out]

((5*a^2*tan(x))/8 + (3*a^2*tan(x)^3)/8)/(tan(x)^2 + 1)^2 + (3*a^2*x)/8

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sympy [B]  time = 1.12, size = 110, normalized size = 3.33 \[ \frac {3 a^{2} x \sin ^{4}{\relax (x )}}{8} + \frac {3 a^{2} x \sin ^{2}{\relax (x )} \cos ^{2}{\relax (x )}}{4} - a^{2} x \sin ^{2}{\relax (x )} + \frac {3 a^{2} x \cos ^{4}{\relax (x )}}{8} - a^{2} x \cos ^{2}{\relax (x )} + a^{2} x - \frac {5 a^{2} \sin ^{3}{\relax (x )} \cos {\relax (x )}}{8} - \frac {3 a^{2} \sin {\relax (x )} \cos ^{3}{\relax (x )}}{8} + a^{2} \sin {\relax (x )} \cos {\relax (x )} \]

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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