Integrand size = 21, antiderivative size = 94 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sin (c+d x)}{a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {4 \sin ^3(c+d x)}{3 a d} \]
-3/2*x/a+4*sin(d*x+c)/a/d-3/2*cos(d*x+c)*sin(d*x+c)/a/d-cos(d*x+c)^3*sin(d *x+c)/d/(a+a*cos(d*x+c))-4/3*sin(d*x+c)^3/a/d
Time = 0.74 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-36 d x \cos \left (\frac {d x}{2}\right )-36 d x \cos \left (c+\frac {d x}{2}\right )+69 \sin \left (\frac {d x}{2}\right )+21 \sin \left (c+\frac {d x}{2}\right )+18 \sin \left (c+\frac {3 d x}{2}\right )+18 \sin \left (2 c+\frac {3 d x}{2}\right )-2 \sin \left (2 c+\frac {5 d x}{2}\right )-2 \sin \left (3 c+\frac {5 d x}{2}\right )+\sin \left (3 c+\frac {7 d x}{2}\right )+\sin \left (4 c+\frac {7 d x}{2}\right )\right )}{48 a d} \]
(Sec[c/2]*Sec[(c + d*x)/2]*(-36*d*x*Cos[(d*x)/2] - 36*d*x*Cos[c + (d*x)/2] + 69*Sin[(d*x)/2] + 21*Sin[c + (d*x)/2] + 18*Sin[c + (3*d*x)/2] + 18*Sin[ 2*c + (3*d*x)/2] - 2*Sin[2*c + (5*d*x)/2] - 2*Sin[3*c + (5*d*x)/2] + Sin[3 *c + (7*d*x)/2] + Sin[4*c + (7*d*x)/2]))/(48*a*d)
Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3246, 3042, 3227, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{a \cos (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 3246 |
| \(\displaystyle -\frac {\int \cos ^2(c+d x) (3 a-4 a \cos (c+d x))dx}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 a-4 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {3 a \int \cos ^2(c+d x)dx-4 a \int \cos ^3(c+d x)dx}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-4 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {\frac {4 a \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}+3 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {4 a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {3 a \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {4 a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {4 a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}+3 a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}\) |
-((Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))) - (3*a*(x/2 + (C os[c + d*x]*Sin[c + d*x])/(2*d)) + (4*a*(-Sin[c + d*x] + Sin[c + d*x]^3/3) )/d)/a^2
3.1.44.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(a + b*Sin[e + f*x]))), x] - Simp[d/(a*b) Int[(c + d* Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[ e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] & & EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2*n] || EqQ[c, 0])
Time = 0.78 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.57
| method | result | size |
| parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (31+\cos \left (3 d x +3 c \right )-\cos \left (2 d x +2 c \right )+17 \cos \left (d x +c \right )\right )-18 d x}{12 a d}\) | \(54\) |
| derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 \left (-\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(85\) |
| default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 \left (-\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(85\) |
| risch | \(-\frac {3 x}{2 a}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )}}{8 a d}+\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {\sin \left (3 d x +3 c \right )}{12 a d}-\frac {\sin \left (2 d x +2 c \right )}{4 a d}\) | \(100\) |
| norman | \(\frac {\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {3 x}{2 a}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {37 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {49 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {9 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {6 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {9 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {6 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(184\) |
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {9 \, d x \cos \left (d x + c\right ) + 9 \, d x - {\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 7 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
-1/6*(9*d*x*cos(d*x + c) + 9*d*x - (2*cos(d*x + c)^3 - cos(d*x + c)^2 + 7* cos(d*x + c) + 16)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)
Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (80) = 160\).
Time = 1.22 (sec) , antiderivative size = 570, normalized size of antiderivative = 6.06 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\begin {cases} - \frac {9 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} - \frac {27 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} - \frac {27 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} - \frac {9 d x}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {6 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {48 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {50 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} + \frac {24 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{a \cos {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
Piecewise((-9*d*x*tan(c/2 + d*x/2)**6/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d* tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 27*d*x*tan(c/2 + d*x/2)**4/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18* a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 27*d*x*tan(c/2 + d*x/2)**2/(6*a*d*tan(c /2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 9*d*x/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 6*tan(c/2 + d*x/2)**7/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6 *a*d) + 48*tan(c/2 + d*x/2)**5/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 50*tan(c/2 + d*x/2)** 3/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 24*tan(c/2 + d*x/2)/(6*a*d*tan(c/2 + d*x/2)**6 + 1 8*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d), Ne(d, 0)) , (x*cos(c)**4/(a*cos(c) + a), True))
Time = 0.49 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.87 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{3 \, d} \]
1/3*((9*sin(d*x + c)/(cos(d*x + c) + 1) + 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a + 3*a*sin(d*x + c)^2/( cos(d*x + c) + 1)^2 + 3*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) - 9*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 3*sin(d*x + c)/(a*(cos(d*x + c) + 1)))/d
Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {9 \, {\left (d x + c\right )}}{a} - \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \]
-1/6*(9*(d*x + c)/a - 6*tan(1/2*d*x + 1/2*c)/a - 2*(15*tan(1/2*d*x + 1/2*c )^5 + 16*tan(1/2*d*x + 1/2*c)^3 + 9*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a))/d
Time = 14.75 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {3\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-\frac {\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{12}+\frac {\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{24}}{a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {3\,x}{2\,a} \]