3.34 \(\int (a-a \sin ^2(x))^4 \, dx\)

Optimal. Leaf size=59 \[ \frac{35 a^4 x}{128}+\frac{1}{8} a^4 \sin (x) \cos ^7(x)+\frac{7}{48} a^4 \sin (x) \cos ^5(x)+\frac{35}{192} a^4 \sin (x) \cos ^3(x)+\frac{35}{128} a^4 \sin (x) \cos (x) \]

[Out]

(35*a^4*x)/128 + (35*a^4*Cos[x]*Sin[x])/128 + (35*a^4*Cos[x]^3*Sin[x])/192 + (7*a^4*Cos[x]^5*Sin[x])/48 + (a^4
*Cos[x]^7*Sin[x])/8

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Rubi [A]  time = 0.0415468, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3175, 2635, 8} \[ \frac{35 a^4 x}{128}+\frac{1}{8} a^4 \sin (x) \cos ^7(x)+\frac{7}{48} a^4 \sin (x) \cos ^5(x)+\frac{35}{192} a^4 \sin (x) \cos ^3(x)+\frac{35}{128} a^4 \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sin[x]^2)^4,x]

[Out]

(35*a^4*x)/128 + (35*a^4*Cos[x]*Sin[x])/128 + (35*a^4*Cos[x]^3*Sin[x])/192 + (7*a^4*Cos[x]^5*Sin[x])/48 + (a^4
*Cos[x]^7*Sin[x])/8

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 )^4 \, dx &=a^4 \int \cos ^8(x) \, dx\\ &=\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{8} \left (7 a^4\right ) \int \cos ^6(x) \, dx\\ &=\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{48} \left (35 a^4\right ) \int \cos ^4(x) \, dx\\ &=\frac{35}{192} a^4 \cos ^3(x) \sin (x)+\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{64} \left (35 a^4\right ) \int \cos ^2(x) \, dx\\ &=\frac{35}{128} a^4 \cos (x) \sin (x)+\frac{35}{192} a^4 \cos ^3(x) \sin (x)+\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{128} \left (35 a^4\right ) \int 1 \, dx\\ &=\frac{35 a^4 x}{128}+\frac{35}{128} a^4 \cos (x) \sin (x)+\frac{35}{192} a^4 \cos ^3(x) \sin (x)+\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0030452, size = 42, normalized size = 0.71 \[ a^4 \left (\frac{35 x}{128}+\frac{7}{32} \sin (2 x)+\frac{7}{128} \sin (4 x)+\frac{1}{96} \sin (6 x)+\frac{\sin (8 x)}{1024}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sin[x]^2)^4,x]

[Out]

a^4*((35*x)/128 + (7*Sin[2*x])/32 + (7*Sin[4*x])/128 + Sin[6*x]/96 + Sin[8*x]/1024)

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Maple [B]  time = 0.02, size = 105, normalized size = 1.8 \begin{align*}{a}^{4} \left ( -{\frac{\cos \left ( x \right ) }{8} \left ( \left ( \sin \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{128}} \right ) -4\,{a}^{4} \left ( -1/6\, \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+5/4\, \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) \cos \left ( x \right ) +{\frac{5\,x}{16}} \right ) +6\,{a}^{4} \left ( -1/4\, \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+3/2\,\sin \left ( x \right ) \right ) \cos \left ( x \right ) +3/8\,x \right ) -4\,{a}^{4} \left ( -1/2\,\sin \left ( x \right ) \cos \left ( x \right ) +x/2 \right ) +{a}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*sin(x)^2)^4,x)

[Out]

a^4*(-1/8*(sin(x)^7+7/6*sin(x)^5+35/24*sin(x)^3+35/16*sin(x))*cos(x)+35/128*x)-4*a^4*(-1/6*(sin(x)^5+5/4*sin(x
)^3+15/8*sin(x))*cos(x)+5/16*x)+6*a^4*(-1/4*(sin(x)^3+3/2*sin(x))*cos(x)+3/8*x)-4*a^4*(-1/2*sin(x)*cos(x)+1/2*
x)+a^4*x

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Maxima [B]  time = 0.958191, size = 140, normalized size = 2.37 \begin{align*} \frac{1}{3072} \,{\left (128 \, \sin \left (2 \, x\right )^{3} + 840 \, x + 3 \, \sin \left (8 \, x\right ) + 168 \, \sin \left (4 \, x\right ) - 768 \, \sin \left (2 \, x\right )\right )} a^{4} - \frac{1}{48} \,{\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{4} + \frac{3}{16} \, a^{4}{\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - a^{4}{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(x)^2)^4,x, algorithm="maxima")

[Out]

1/3072*(128*sin(2*x)^3 + 840*x + 3*sin(8*x) + 168*sin(4*x) - 768*sin(2*x))*a^4 - 1/48*(4*sin(2*x)^3 + 60*x + 9
*sin(4*x) - 48*sin(2*x))*a^4 + 3/16*a^4*(12*x + sin(4*x) - 8*sin(2*x)) - a^4*(2*x - sin(2*x)) + a^4*x

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Fricas [A]  time = 1.74637, size = 135, normalized size = 2.29 \begin{align*} \frac{35}{128} \, a^{4} x + \frac{1}{384} \,{\left (48 \, a^{4} \cos \left (x\right )^{7} + 56 \, a^{4} \cos \left (x\right )^{5} + 70 \, a^{4} \cos \left (x\right )^{3} + 105 \, a^{4} \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)^4,x, algorithm="fricas")

[Out]

35/128*a^4*x + 1/384*(48*a^4*cos(x)^7 + 56*a^4*cos(x)^5 + 70*a^4*cos(x)^3 + 105*a^4*cos(x))*sin(x)

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Sympy [B]  time = 13.8102, size = 376, normalized size = 6.37 \begin{align*} \frac{35 a^{4} x \sin ^{8}{\left (x \right )}}{128} + \frac{35 a^{4} x \sin ^{6}{\left (x \right )} \cos ^{2}{\left (x \right )}}{32} - \frac{5 a^{4} x \sin ^{6}{\left (x \right )}}{4} + \frac{105 a^{4} x \sin ^{4}{\left (x \right )} \cos ^{4}{\left (x \right )}}{64} - \frac{15 a^{4} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac{9 a^{4} x \sin ^{4}{\left (x \right )}}{4} + \frac{35 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{6}{\left (x \right )}}{32} - \frac{15 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{4} + \frac{9 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} - 2 a^{4} x \sin ^{2}{\left (x \right )} + \frac{35 a^{4} x \cos ^{8}{\left (x \right )}}{128} - \frac{5 a^{4} x \cos ^{6}{\left (x \right )}}{4} + \frac{9 a^{4} x \cos ^{4}{\left (x \right )}}{4} - 2 a^{4} x \cos ^{2}{\left (x \right )} + a^{4} x - \frac{93 a^{4} \sin ^{7}{\left (x \right )} \cos{\left (x \right )}}{128} - \frac{511 a^{4} \sin ^{5}{\left (x \right )} \cos ^{3}{\left (x \right )}}{384} + \frac{11 a^{4} \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{4} - \frac{385 a^{4} \sin ^{3}{\left (x \right )} \cos ^{5}{\left (x \right )}}{384} + \frac{10 a^{4} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{3} - \frac{15 a^{4} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{4} - \frac{35 a^{4} \sin{\left (x \right )} \cos ^{7}{\left (x \right )}}{128} + \frac{5 a^{4} \sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{4} - \frac{9 a^{4} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + 2 a^{4} \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)**4,x)

[Out]

35*a**4*x*sin(x)**8/128 + 35*a**4*x*sin(x)**6*cos(x)**2/32 - 5*a**4*x*sin(x)**6/4 + 105*a**4*x*sin(x)**4*cos(x
)**4/64 - 15*a**4*x*sin(x)**4*cos(x)**2/4 + 9*a**4*x*sin(x)**4/4 + 35*a**4*x*sin(x)**2*cos(x)**6/32 - 15*a**4*
x*sin(x)**2*cos(x)**4/4 + 9*a**4*x*sin(x)**2*cos(x)**2/2 - 2*a**4*x*sin(x)**2 + 35*a**4*x*cos(x)**8/128 - 5*a*
*4*x*cos(x)**6/4 + 9*a**4*x*cos(x)**4/4 - 2*a**4*x*cos(x)**2 + a**4*x - 93*a**4*sin(x)**7*cos(x)/128 - 511*a**
4*sin(x)**5*cos(x)**3/384 + 11*a**4*sin(x)**5*cos(x)/4 - 385*a**4*sin(x)**3*cos(x)**5/384 + 10*a**4*sin(x)**3*
cos(x)**3/3 - 15*a**4*sin(x)**3*cos(x)/4 - 35*a**4*sin(x)*cos(x)**7/128 + 5*a**4*sin(x)*cos(x)**5/4 - 9*a**4*s
in(x)*cos(x)**3/4 + 2*a**4*sin(x)*cos(x)

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Giac [A]  time = 1.13789, size = 58, normalized size = 0.98 \begin{align*} \frac{35}{128} \, a^{4} x + \frac{1}{1024} \, a^{4} \sin \left (8 \, x\right ) + \frac{1}{96} \, a^{4} \sin \left (6 \, x\right ) + \frac{7}{128} \, a^{4} \sin \left (4 \, x\right ) + \frac{7}{32} \, a^{4} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(x)^2)^4,x, algorithm="giac")

[Out]

35/128*a^4*x + 1/1024*a^4*sin(8*x) + 1/96*a^4*sin(6*x) + 7/128*a^4*sin(4*x) + 7/32*a^4*sin(2*x)