Integrand size = 19, antiderivative size = 102 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {7 a^4 x}{2}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {7 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}-\frac {8 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \] Output:
7/2*a^4*x+8*a^4*sin(d*x+c)/d+7/2*a^4*cos(d*x+c)*sin(d*x+c)/d+a^4*cos(d*x+c )^3*sin(d*x+c)/d-8/3*a^4*sin(d*x+c)^3/d+1/5*a^4*sin(d*x+c)^5/d
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {a^4 (840 d x+1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x)))}{240 d} \] Input:
Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^4,x]
Output:
(a^4*(840*d*x + 1470*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 145*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] + 3*Sin[5*(c + d*x)]))/(240*d)
Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 3230, 3042, 3124, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle \frac {4}{5} \int (\cos (c+d x) a+a)^4dx+\frac {\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4}{5} \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx+\frac {\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
\(\Big \downarrow \) 3124 |
\(\displaystyle \frac {4}{5} \int \left (\cos ^4(c+d x) a^4+4 \cos ^3(c+d x) a^4+6 \cos ^2(c+d x) a^4+4 \cos (c+d x) a^4+a^4\right )dx+\frac {\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{5} \left (-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\right )+\frac {\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
Input:
Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^4,x]
Output:
((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + (4*((35*a^4*x)/8 + (8*a^4*Si n[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d* x]^3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/(3*d)))/5
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 33.60 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(\frac {a^{4} \left (840 d x +1470 \sin \left (d x +c \right )+3 \sin \left (5 d x +5 c \right )+30 \sin \left (4 d x +4 c \right )+145 \sin \left (3 d x +3 c \right )+480 \sin \left (2 d x +2 c \right )\right )}{240 d}\) | \(66\) |
risch | \(\frac {7 a^{4} x}{2}+\frac {49 a^{4} \sin \left (d x +c \right )}{8 d}+\frac {a^{4} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{8 d}+\frac {29 a^{4} \sin \left (3 d x +3 c \right )}{48 d}+\frac {2 a^{4} \sin \left (2 d x +2 c \right )}{d}\) | \(90\) |
derivativedivides | \(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+4 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \sin \left (d x +c \right )}{d}\) | \(133\) |
default | \(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+4 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \sin \left (d x +c \right )}{d}\) | \(133\) |
parts | \(\frac {a^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {4 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{d}+\frac {4 a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(144\) |
norman | \(\frac {\frac {7 a^{4} x}{2}+\frac {25 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {158 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {896 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {98 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {7 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {35 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+35 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+35 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {35 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {7 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(202\) |
orering | \(\text {Expression too large to display}\) | \(2147\) |
Input:
int(cos(d*x+c)*(a+a*cos(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/240*a^4*(840*d*x+1470*sin(d*x+c)+3*sin(5*d*x+5*c)+30*sin(4*d*x+4*c)+145* sin(3*d*x+3*c)+480*sin(2*d*x+2*c))/d
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {105 \, a^{4} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{4} + 30 \, a^{4} \cos \left (d x + c\right )^{3} + 68 \, a^{4} \cos \left (d x + c\right )^{2} + 105 \, a^{4} \cos \left (d x + c\right ) + 166 \, a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4,x, algorithm="fricas")
Output:
1/30*(105*a^4*d*x + (6*a^4*cos(d*x + c)^4 + 30*a^4*cos(d*x + c)^3 + 68*a^4 *cos(d*x + c)^2 + 105*a^4*cos(d*x + c) + 166*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (95) = 190\).
Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.75 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\begin {cases} \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{4} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))**4,x)
Output:
Piecewise((3*a**4*x*sin(c + d*x)**4/2 + 3*a**4*x*sin(c + d*x)**2*cos(c + d *x)**2 + 2*a**4*x*sin(c + d*x)**2 + 3*a**4*x*cos(c + d*x)**4/2 + 2*a**4*x* cos(c + d*x)**2 + 8*a**4*sin(c + d*x)**5/(15*d) + 4*a**4*sin(c + d*x)**3*c os(c + d*x)**2/(3*d) + 3*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 4*a**4* sin(c + d*x)**3/d + a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*a**4*sin(c + d *x)*cos(c + d*x)**3/(2*d) + 6*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 2*a**4 *sin(c + d*x)*cos(c + d*x)/d + a**4*sin(c + d*x)/d, Ne(d, 0)), (x*(a*cos(c ) + a)**4*cos(c), True))
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.25 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {8 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, a^{4} \sin \left (d x + c\right )}{120 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4,x, algorithm="maxima")
Output:
1/120*(8*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^4 - 24 0*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^4 + 120*a^4*sin(d*x + c))/d
Time = 0.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {7}{2} \, a^{4} x + \frac {a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{8 \, d} + \frac {29 \, a^{4} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {2 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac {49 \, a^{4} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4,x, algorithm="giac")
Output:
7/2*a^4*x + 1/80*a^4*sin(5*d*x + 5*c)/d + 1/8*a^4*sin(4*d*x + 4*c)/d + 29/ 48*a^4*sin(3*d*x + 3*c)/d + 2*a^4*sin(2*d*x + 2*c)/d + 49/8*a^4*sin(d*x + c)/d
Time = 47.01 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.03 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {7\,a^4\,x}{2}+\frac {7\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {98\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {896\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\frac {158\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \] Input:
int(cos(c + d*x)*(a + a*cos(c + d*x))^4,x)
Output:
(7*a^4*x)/2 + ((158*a^4*tan(c/2 + (d*x)/2)^3)/3 + (896*a^4*tan(c/2 + (d*x) /2)^5)/15 + (98*a^4*tan(c/2 + (d*x)/2)^7)/3 + 7*a^4*tan(c/2 + (d*x)/2)^9 + 25*a^4*tan(c/2 + (d*x)/2))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^5)
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx=\frac {a^{4} \left (-30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+135 \cos \left (d x +c \right ) \sin \left (d x +c \right )+6 \sin \left (d x +c \right )^{5}-80 \sin \left (d x +c \right )^{3}+240 \sin \left (d x +c \right )+105 d x \right )}{30 d} \] Input:
int(cos(d*x+c)*(a+a*cos(d*x+c))^4,x)
Output:
(a**4*( - 30*cos(c + d*x)*sin(c + d*x)**3 + 135*cos(c + d*x)*sin(c + d*x) + 6*sin(c + d*x)**5 - 80*sin(c + d*x)**3 + 240*sin(c + d*x) + 105*d*x))/(3 0*d)