3.4.64 \(\int (4-3 \cos (x)) (1-\frac {\sin (x)}{2})^4 \, dx\) [364]

Optimal. Leaf size=70 \[ \frac {227 x}{32}+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80} \]

[Out]

227/32*x+10*cos(x)-3*cos(x)^2-2/3*cos(x)^3-3*sin(x)-99/32*cos(x)*sin(x)-3/2*sin(x)^3-1/16*cos(x)*sin(x)^3+3/8*
sin(x)^4-3/80*sin(x)^5

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Rubi [A]
time = 0.11, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4486, 2717, 2747, 2748, 2715, 8, 655} \begin {gather*} \frac {227 x}{32}-\frac {3}{80} \sin ^5(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^3(x)}{2}-3 \sin (x)-\frac {2 \cos ^3(x)}{3}-3 \cos ^2(x)+10 \cos (x)-\frac {1}{16} \sin ^3(x) \cos (x)-\frac {99}{32} \sin (x) \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 3*Cos[x])*(1 - Sin[x]/2)^4,x]

[Out]

(227*x)/32 + 10*Cos[x] - 3*Cos[x]^2 - (2*Cos[x]^3)/3 - 3*Sin[x] - (99*Cos[x]*Sin[x])/32 - (3*Sin[x]^3)/2 - (Co
s[x]*Sin[x]^3)/16 + (3*Sin[x]^4)/8 - (3*Sin[x]^5)/80

Rule 8

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

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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 2717

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

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 4486

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rubi steps

\begin {align*} \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx &=\int \left (4-3 \cos (x)+2 (-4+3 \cos (x)) \sin (x)-\frac {3}{2} (-4+3 \cos (x)) \sin ^2(x)+\frac {1}{2} (-4+3 \cos (x)) \sin ^3(x)-\frac {1}{16} (-4+3 \cos (x)) \sin ^4(x)\right ) \, dx\\ &=4 x-\frac {1}{16} \int (-4+3 \cos (x)) \sin ^4(x) \, dx+\frac {1}{2} \int (-4+3 \cos (x)) \sin ^3(x) \, dx-\frac {3}{2} \int (-4+3 \cos (x)) \sin ^2(x) \, dx+2 \int (-4+3 \cos (x)) \sin (x) \, dx-3 \int \cos (x) \, dx\\ &=4 x-3 \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {3 \sin ^5(x)}{80}-\frac {1}{54} \text {Subst}\left (\int (-4+x) \left (9-x^2\right ) \, dx,x,3 \cos (x)\right )+\frac {1}{4} \int \sin ^4(x) \, dx-\frac {2}{3} \text {Subst}(\int (-4+x) \, dx,x,3 \cos (x))+6 \int \sin ^2(x) \, dx\\ &=4 x+8 \cos (x)-3 \cos ^2(x)-3 \sin (x)-3 \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80}+\frac {2}{27} \text {Subst}\left (\int \left (9-x^2\right ) \, dx,x,3 \cos (x)\right )+\frac {3}{16} \int \sin ^2(x) \, dx+3 \int 1 \, dx\\ &=7 x+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80}+\frac {3 \int 1 \, dx}{32}\\ &=\frac {227 x}{32}+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 74, normalized size = 1.06 \begin {gather*} \frac {227 x}{32}+\frac {19 \cos (x)}{2}-\frac {27}{16} \cos (2 x)-\frac {1}{6} \cos (3 x)+\frac {3}{64} \cos (4 x)-\frac {531 \sin (x)}{128}-\frac {25}{16} \sin (2 x)+\frac {99}{256} \sin (3 x)+\frac {1}{128} \sin (4 x)-\frac {3 \sin (5 x)}{1280} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 3*Cos[x])*(1 - Sin[x]/2)^4,x]

[Out]

(227*x)/32 + (19*Cos[x])/2 - (27*Cos[2*x])/16 - Cos[3*x]/6 + (3*Cos[4*x])/64 - (531*Sin[x])/128 - (25*Sin[2*x]
)/16 + (99*Sin[3*x])/256 + Sin[4*x]/128 - (3*Sin[5*x])/1280

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Maple [A]
time = 0.15, size = 66, normalized size = 0.94

method	result	size
risch	\(\frac {227 x}{32}+\frac {19 \cos \left (x \right )}{2}-\frac {531 \sin \left (x \right )}{128}-\frac {3 \sin \left (5 x \right )}{1280}+\frac {3 \cos \left (4 x \right )}{64}+\frac {\sin \left (4 x \right )}{128}-\frac {\cos \left (3 x \right )}{6}+\frac {99 \sin \left (3 x \right )}{256}-\frac {27 \cos \left (2 x \right )}{16}-\frac {25 \sin \left (2 x \right )}{16}\)	\(55\)
default	\(\frac {227 x}{32}+8 \cos \left (x \right )-3 \cos \left (x \right ) \sin \left (x \right )+\frac {2 \left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{3}-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{16}-3 \sin \left (x \right )-3 \left (\cos ^{2}\left (x \right )\right )-\frac {3 \left (\sin ^{3}\left (x \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (x \right )\right )}{8}-\frac {3 \left (\sin ^{5}\left (x \right )\right )}{80}\)	\(66\)
norman	\(\frac {11 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {151 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{3}-\frac {128 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3}-\frac {47 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{3}+\frac {227 x}{32}-\frac {391 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{8}-\frac {306 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {185 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{16}+\frac {1135 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{32}+\frac {1135 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {1135 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {1135 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{32}+\frac {227 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{32}-\frac {195 \tan \left (\frac {x}{2}\right )}{16}+3}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5}}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4-3*cos(x))*(1-1/2*sin(x))^4,x,method=_RETURNVERBOSE)

[Out]

227/32*x+8*cos(x)-3*cos(x)*sin(x)+2/3*(2+sin(x)^2)*cos(x)-1/16*(sin(x)^3+3/2*sin(x))*cos(x)-3*sin(x)-3*cos(x)^
2-3/2*sin(x)^3+3/8*sin(x)^4-3/80*sin(x)^5

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Maxima [A]
time = 2.63, size = 54, normalized size = 0.77 \begin {gather*} -\frac {3}{80} \, \sin \left (x\right )^{5} + \frac {3}{8} \, \sin \left (x\right )^{4} - \frac {2}{3} \, \cos \left (x\right )^{3} - \frac {3}{2} \, \sin \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \frac {227}{32} \, x + 10 \, \cos \left (x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) - \frac {25}{16} \, \sin \left (2 \, x\right ) - 3 \, \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*cos(x))*(1-1/2*sin(x))^4,x, algorithm="maxima")

[Out]

-3/80*sin(x)^5 + 3/8*sin(x)^4 - 2/3*cos(x)^3 - 3/2*sin(x)^3 - 3*cos(x)^2 + 227/32*x + 10*cos(x) + 1/128*sin(4*
x) - 25/16*sin(2*x) - 3*sin(x)

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Fricas [A]
time = 0.93, size = 54, normalized size = 0.77 \begin {gather*} \frac {3}{8} \, \cos \left (x\right )^{4} - \frac {2}{3} \, \cos \left (x\right )^{3} - \frac {15}{4} \, \cos \left (x\right )^{2} - \frac {1}{160} \, {\left (6 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{3} - 252 \, \cos \left (x\right )^{2} + 505 \, \cos \left (x\right ) + 726\right )} \sin \left (x\right ) + \frac {227}{32} \, x + 10 \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*cos(x))*(1-1/2*sin(x))^4,x, algorithm="fricas")

[Out]

3/8*cos(x)^4 - 2/3*cos(x)^3 - 15/4*cos(x)^2 - 1/160*(6*cos(x)^4 - 10*cos(x)^3 - 252*cos(x)^2 + 505*cos(x) + 72
6)*sin(x) + 227/32*x + 10*cos(x)

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Sympy [A]
time = 0.25, size = 148, normalized size = 2.11 \begin {gather*} \frac {3 x \sin ^{4}{\left (x \right )}}{32} + \frac {3 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + 3 x \sin ^{2}{\left (x \right )} + \frac {3 x \cos ^{4}{\left (x \right )}}{32} + 3 x \cos ^{2}{\left (x \right )} + 4 x - \frac {3 \sin ^{5}{\left (x \right )}}{80} + \frac {3 \sin ^{4}{\left (x \right )}}{8} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{32} - \frac {3 \sin ^{3}{\left (x \right )}}{2} + 2 \sin ^{2}{\left (x \right )} \cos {\left (x \right )} + 3 \sin ^{2}{\left (x \right )} - \frac {3 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{32} - 3 \sin {\left (x \right )} \cos {\left (x \right )} - 3 \sin {\left (x \right )} + \frac {4 \cos ^{3}{\left (x \right )}}{3} + 8 \cos {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*cos(x))*(1-1/2*sin(x))**4,x)

[Out]

3*x*sin(x)**4/32 + 3*x*sin(x)**2*cos(x)**2/16 + 3*x*sin(x)**2 + 3*x*cos(x)**4/32 + 3*x*cos(x)**2 + 4*x - 3*sin
(x)**5/80 + 3*sin(x)**4/8 - 5*sin(x)**3*cos(x)/32 - 3*sin(x)**3/2 + 2*sin(x)**2*cos(x) + 3*sin(x)**2 - 3*sin(x
)*cos(x)**3/32 - 3*sin(x)*cos(x) - 3*sin(x) + 4*cos(x)**3/3 + 8*cos(x)

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Giac [A]
time = 1.34, size = 54, normalized size = 0.77 \begin {gather*} \frac {227}{32} \, x + \frac {3}{64} \, \cos \left (4 \, x\right ) - \frac {1}{6} \, \cos \left (3 \, x\right ) - \frac {27}{16} \, \cos \left (2 \, x\right ) + \frac {19}{2} \, \cos \left (x\right ) - \frac {3}{1280} \, \sin \left (5 \, x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) + \frac {99}{256} \, \sin \left (3 \, x\right ) - \frac {25}{16} \, \sin \left (2 \, x\right ) - \frac {531}{128} \, \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*cos(x))*(1-1/2*sin(x))^4,x, algorithm="giac")

[Out]

227/32*x + 3/64*cos(4*x) - 1/6*cos(3*x) - 27/16*cos(2*x) + 19/2*cos(x) - 3/1280*sin(5*x) + 1/128*sin(4*x) + 99
/256*sin(3*x) - 25/16*sin(2*x) - 531/128*sin(x)

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Mupad [B]
time = 0.38, size = 94, normalized size = 1.34 \begin {gather*} -\frac {6\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^9}{5}+6\,{\cos \left (\frac {x}{2}\right )}^8+\frac {17\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^7}{5}-\frac {52\,{\cos \left (\frac {x}{2}\right )}^6}{3}+\frac {93\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^5}{10}+2\,{\cos \left (\frac {x}{2}\right )}^4-\frac {191\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^3}{8}+28\,{\cos \left (\frac {x}{2}\right )}^2+\frac {3\,\sin \left (\frac {x}{2}\right )\,\cos \left (\frac {x}{2}\right )}{16}+\frac {227\,x}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*cos(x) - 4)*(sin(x)/2 - 1)^4,x)

[Out]

(227*x)/32 - (191*cos(x/2)^3*sin(x/2))/8 + (93*cos(x/2)^5*sin(x/2))/10 + (17*cos(x/2)^7*sin(x/2))/5 - (6*cos(x
/2)^9*sin(x/2))/5 + 28*cos(x/2)^2 + 2*cos(x/2)^4 - (52*cos(x/2)^6)/3 + 6*cos(x/2)^8 + (3*cos(x/2)*sin(x/2))/16

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