3.1.34 \(\int (a-a \sin ^2(x))^4 \, dx\) [34]

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

[Out]

35/128*a^4*x+35/128*a^4*cos(x)*sin(x)+35/192*a^4*cos(x)^3*sin(x)+7/48*a^4*cos(x)^5*sin(x)+1/8*a^4*cos(x)^7*sin
(x)

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 = {3254, 2715, 8} \begin {gather*} \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) \end {gather*}

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 8

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

Rule 2715

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] && Integ
erQ[2*n]

Rule 3254

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 )^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*}

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Mathematica [A]
time = 0.00, size = 42, normalized size = 0.71 \begin {gather*} 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 ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(49)=98\).
time = 0.23, size = 105, normalized size = 1.78

method	result	size
risch	\(\frac {35 a^{4} x}{128}+\frac {a^{4} \sin \left (8 x \right )}{1024}+\frac {a^{4} \sin \left (6 x \right )}{96}+\frac {7 a^{4} \sin \left (4 x \right )}{128}+\frac {7 a^{4} \sin \left (2 x \right )}{32}\)	\(44\)
default	\(a^{4} \left (-\frac {\left (\sin ^{7}\left (x \right )+\frac {7 \left (\sin ^{5}\left (x \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (x \right )\right )}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {35 x}{128}\right )-4 a^{4} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+6 a^{4} \left (-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-4 a^{4} \left (-\frac {\sin \left (x \right ) \cos \left (x \right )}{2}+\frac {x}{2}\right )+a^{4} x\)	\(105\)
norman	\(\frac {\frac {35 a^{4} x}{128}+\frac {93 a^{4} \tan \left (\frac {x}{2}\right )}{64}+\frac {91 a^{4} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{192}+\frac {1799 a^{4} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{192}-\frac {1085 a^{4} \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{192}+\frac {1085 a^{4} \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{192}-\frac {1799 a^{4} \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{192}-\frac {91 a^{4} \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{192}-\frac {93 a^{4} \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{64}+\frac {35 a^{4} x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{16}+\frac {245 a^{4} x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{32}+\frac {245 a^{4} x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {1225 a^{4} x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{64}+\frac {245 a^{4} x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{16}+\frac {245 a^{4} x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{32}+\frac {35 a^{4} x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{16}+\frac {35 a^{4} x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{128}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{8}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (49) = 98\).
time = 0.30, size = 104, normalized size = 1.76 \begin {gather*} \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 {gather*}

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 = 0.39, size = 46, normalized size = 0.78 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (65) = 130\).
time = 0.71, size = 376, normalized size = 6.37 \begin {gather*} \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 {gather*}

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 = 0.44, size = 43, normalized size = 0.73 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 13.69, size = 51, normalized size = 0.86 \begin {gather*} \frac {\frac {35\,a^4\,{\mathrm {tan}\left (x\right )}^7}{128}+\frac {385\,a^4\,{\mathrm {tan}\left (x\right )}^5}{384}+\frac {511\,a^4\,{\mathrm {tan}\left (x\right )}^3}{384}+\frac {93\,a^4\,\mathrm {tan}\left (x\right )}{128}}{{\left ({\mathrm {tan}\left (x\right )}^2+1\right )}^4}+\frac {35\,a^4\,x}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*sin(x)^2)^4,x)

[Out]

((93*a^4*tan(x))/128 + (511*a^4*tan(x)^3)/384 + (385*a^4*tan(x)^5)/384 + (35*a^4*tan(x)^7)/128)/(tan(x)^2 + 1)
^4 + (35*a^4*x)/128

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