\(\int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx\) [14]

Optimal result

Integrand size = 21, antiderivative size = 103 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {3 a^2 x}{4}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin ^5(c+d x)}{5 d} \]

[Out]

3/4*a^2*x+2*a^2*sin(d*x+c)/d+3/4*a^2*cos(d*x+c)*sin(d*x+c)/d+1/2*a^2*cos(d*x+c)^3*sin(d*x+c)/d-a^2*sin(d*x+c)^
3/d+1/5*a^2*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2836, 2713, 2715, 8} \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {3 a^2 x}{4} \]

[In]

Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2,x]

[Out]

(3*a^2*x)/4 + (2*a^2*Sin[c + d*x])/d + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]^3*Sin[c + d
*x])/(2*d) - (a^2*Sin[c + d*x]^3)/d + (a^2*Sin[c + d*x]^5)/(5*d)

Rule 8

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

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 2836

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cos ^3(c+d x)+2 a^2 \cos ^4(c+d x)+a^2 \cos ^5(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^3(c+d x) \, dx+a^2 \int \cos ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx \\ & = \frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin ^5(c+d x)}{5 d}+\frac {1}{4} \left (3 a^2\right ) \int 1 \, dx \\ & = \frac {3 a^2 x}{4}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin ^5(c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.59 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^2 (60 d x+110 \sin (c+d x)+40 \sin (2 (c+d x))+15 \sin (3 (c+d x))+5 \sin (4 (c+d x))+\sin (5 (c+d x)))}{80 d} \]

[In]

Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2,x]

[Out]

(a^2*(60*d*x + 110*Sin[c + d*x] + 40*Sin[2*(c + d*x)] + 15*Sin[3*(c + d*x)] + 5*Sin[4*(c + d*x)] + Sin[5*(c +
d*x)]))/(80*d)

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.62

method	result	size
parallelrisch	\(\frac {a^{2} \left (60 d x +110 \sin \left (d x +c \right )+\sin \left (5 d x +5 c \right )+5 \sin \left (4 d x +4 c \right )+15 \sin \left (3 d x +3 c \right )+40 \sin \left (2 d x +2 c \right )\right )}{80 d}\)	\(64\)
risch	\(\frac {3 a^{2} x}{4}+\frac {11 a^{2} \sin \left (d x +c \right )}{8 d}+\frac {a^{2} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{16 d}+\frac {3 a^{2} \sin \left (3 d x +3 c \right )}{16 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{2 d}\)	\(90\)
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(96\)
default	\(\frac {\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(96\)
parts	\(\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(101\)
norman	\(\frac {\frac {3 a^{2} x}{4}+\frac {13 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {9 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {72 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {7 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {15 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {15 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {15 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {3 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\)	\(202\)

[In]

int(cos(d*x+c)^3*(a+cos(d*x+c)*a)^2,x,method=_RETURNVERBOSE)

[Out]

1/80*a^2*(60*d*x+110*sin(d*x+c)+sin(5*d*x+5*c)+5*sin(4*d*x+4*c)+15*sin(3*d*x+3*c)+40*sin(2*d*x+2*c))/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {15 \, a^{2} d x + {\left (4 \, a^{2} \cos \left (d x + c\right )^{4} + 10 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 15 \, a^{2} \cos \left (d x + c\right ) + 24 \, a^{2}\right )} \sin \left (d x + c\right )}{20 \, d} \]

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^2,x, algorithm="fricas")

[Out]

1/20*(15*a^2*d*x + (4*a^2*cos(d*x + c)^4 + 10*a^2*cos(d*x + c)^3 + 12*a^2*cos(d*x + c)^2 + 15*a^2*cos(d*x + c)
+ 24*a^2)*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (94) = 188\).

Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.15 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\begin {cases} \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {8 a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{2} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**3*(a+a*cos(d*x+c))**2,x)

[Out]

Piecewise((3*a**2*x*sin(c + d*x)**4/4 + 3*a**2*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + 3*a**2*x*cos(c + d*x)**4/
4 + 8*a**2*sin(c + d*x)**5/(15*d) + 4*a**2*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*a**2*sin(c + d*x)**3*cos(
c + d*x)/(4*d) + 2*a**2*sin(c + d*x)**3/(3*d) + a**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*a**2*sin(c + d*x)*cos(
c + d*x)**3/(4*d) + a**2*sin(c + d*x)*cos(c + d*x)**2/d, Ne(d, 0)), (x*(a*cos(c) + a)**2*cos(c)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.92 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {16 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} - 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{240 \, d} \]

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^2,x, algorithm="maxima")

[Out]

1/240*(16*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^2 - 80*(sin(d*x + c)^3 - 3*sin(d*x + c))*
a^2 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {3}{4} \, a^{2} x + \frac {a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {3 \, a^{2} \sin \left (3 \, d x + 3 \, c\right )}{16 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {11 \, a^{2} \sin \left (d x + c\right )}{8 \, d} \]

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^2,x, algorithm="giac")

[Out]

3/4*a^2*x + 1/80*a^2*sin(5*d*x + 5*c)/d + 1/16*a^2*sin(4*d*x + 4*c)/d + 3/16*a^2*sin(3*d*x + 3*c)/d + 1/2*a^2*
sin(2*d*x + 2*c)/d + 11/8*a^2*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 17.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {3\,a^2\,x}{4}+\frac {\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}+7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {72\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {13\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]

[In]

int(cos(c + d*x)^3*(a + a*cos(c + d*x))^2,x)

[Out]

(3*a^2*x)/4 + (9*a^2*tan(c/2 + (d*x)/2)^3 + (72*a^2*tan(c/2 + (d*x)/2)^5)/5 + 7*a^2*tan(c/2 + (d*x)/2)^7 + (3*
a^2*tan(c/2 + (d*x)/2)^9)/2 + (13*a^2*tan(c/2 + (d*x)/2))/2)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^5)