Integrand size = 12, antiderivative size = 87 \[ \int (a+a \cos (c+d x))^4 \, dx=\frac {35 a^4 x}{8}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d} \]
35/8*a^4*x+8*a^4*sin(d*x+c)/d+27/8*a^4*cos(d*x+c)*sin(d*x+c)/d+1/4*a^4*cos (d*x+c)^3*sin(d*x+c)/d-4/3*a^4*sin(d*x+c)^3/d
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int (a+a \cos (c+d x))^4 \, dx=\frac {a^4 (420 c+420 d x+672 \sin (c+d x)+168 \sin (2 (c+d x))+32 \sin (3 (c+d x))+3 \sin (4 (c+d x)))}{96 d} \]
(a^4*(420*c + 420*d*x + 672*Sin[c + d*x] + 168*Sin[2*(c + d*x)] + 32*Sin[3 *(c + d*x)] + 3*Sin[4*(c + d*x)]))/(96*d)
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.
\(\displaystyle \int (a \cos (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4dx\) |
\(\Big \downarrow \) 3124 |
\(\displaystyle \int \left (a^4 \cos ^4(c+d x)+4 a^4 \cos ^3(c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos (c+d x)+a^4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\) |
(35*a^4*x)/8 + (8*a^4*Sin[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)
3.1.35.3.1 Defintions of rubi rules used
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]
Time = 2.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {a^{4} \left (420 d x +672 \sin \left (d x +c \right )+3 \sin \left (4 d x +4 c \right )+32 \sin \left (3 d x +3 c \right )+168 \sin \left (2 d x +2 c \right )\right )}{96 d}\) | \(55\) |
risch | \(\frac {35 a^{4} x}{8}+\frac {7 a^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} \sin \left (3 d x +3 c \right )}{3 d}+\frac {7 a^{4} \sin \left (2 d x +2 c \right )}{4 d}\) | \(73\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} \sin \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(111\) |
default | \(\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} \sin \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(111\) |
parts | \(a^{4} x +\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {6 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {4 a^{4} \sin \left (d x +c \right )}{d}\) | \(115\) |
norman | \(\frac {\frac {35 a^{4} x}{8}+\frac {93 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {511 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {385 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {35 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {105 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {35 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(166\) |
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int (a+a \cos (c+d x))^4 \, dx=\frac {105 \, a^{4} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} + 32 \, a^{4} \cos \left (d x + c\right )^{2} + 81 \, a^{4} \cos \left (d x + c\right ) + 160 \, a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(105*a^4*d*x + (6*a^4*cos(d*x + c)^3 + 32*a^4*cos(d*x + c)^2 + 81*a^4 *cos(d*x + c) + 160*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (82) = 164\).
Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.57 \[ \int (a+a \cos (c+d x))^4 \, dx=\begin {cases} \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 a^{4} x \cos ^{2}{\left (c + d x \right )} + a^{4} x + \frac {3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {8 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {4 a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((3*a**4*x*sin(c + d*x)**4/8 + 3*a**4*x*sin(c + d*x)**2*cos(c + d *x)**2/4 + 3*a**4*x*sin(c + d*x)**2 + 3*a**4*x*cos(c + d*x)**4/8 + 3*a**4* x*cos(c + d*x)**2 + a**4*x + 3*a**4*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 8 *a**4*sin(c + d*x)**3/(3*d) + 5*a**4*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 4*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*a**4*sin(c + d*x)*cos(c + d*x)/d + 4*a**4*sin(c + d*x)/d, Ne(d, 0)), (x*(a*cos(c) + a)**4, True))
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22 \[ \int (a+a \cos (c+d x))^4 \, dx=a^{4} x - \frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{2 \, d} + \frac {4 \, a^{4} \sin \left (d x + c\right )}{d} \]
a^4*x - 4/3*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^4/d + 1/32*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4/d + 3/2*(2*d*x + 2*c + sin(2* d*x + 2*c))*a^4/d + 4*a^4*sin(d*x + c)/d
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83 \[ \int (a+a \cos (c+d x))^4 \, dx=\frac {35}{8} \, a^{4} x + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a^{4} \sin \left (3 \, d x + 3 \, c\right )}{3 \, d} + \frac {7 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {7 \, a^{4} \sin \left (d x + c\right )}{d} \]
35/8*a^4*x + 1/32*a^4*sin(4*d*x + 4*c)/d + 1/3*a^4*sin(3*d*x + 3*c)/d + 7/ 4*a^4*sin(2*d*x + 2*c)/d + 7*a^4*sin(d*x + c)/d
Time = 17.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int (a+a \cos (c+d x))^4 \, dx=\frac {35\,a^4\,x}{8}+\frac {\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {385\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {511\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}+\frac {93\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]