Integrand size = 21, antiderivative size = 76 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {3 x}{2 a}-\frac {2 \sin (c+d x)}{a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\cos ^2(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))} \]
3/2*x/a-2*sin(d*x+c)/a/d+3/2*cos(d*x+c)*sin(d*x+c)/a/d-cos(d*x+c)^2*sin(d* x+c)/d/(a+a*cos(d*x+c))
Time = 0.72 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.54 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (12 d x \cos \left (\frac {d x}{2}\right )+12 d x \cos \left (c+\frac {d x}{2}\right )-20 \sin \left (\frac {d x}{2}\right )-4 \sin \left (c+\frac {d x}{2}\right )-3 \sin \left (c+\frac {3 d x}{2}\right )-3 \sin \left (2 c+\frac {3 d x}{2}\right )+\sin \left (2 c+\frac {5 d x}{2}\right )+\sin \left (3 c+\frac {5 d x}{2}\right )\right )}{16 a d} \]
(Sec[c/2]*Sec[(c + d*x)/2]*(12*d*x*Cos[(d*x)/2] + 12*d*x*Cos[c + (d*x)/2] - 20*Sin[(d*x)/2] - 4*Sin[c + (d*x)/2] - 3*Sin[c + (3*d*x)/2] - 3*Sin[2*c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2] + Sin[3*c + (5*d*x)/2]))/(16*a*d)
Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3246, 3042, 3213}
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.
\(\displaystyle \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 3246 |
\(\displaystyle -\frac {\int \cos (c+d x) (2 a-3 a \cos (c+d x))dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 a-3 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 3213 |
\(\displaystyle -\frac {\frac {2 a \sin (c+d x)}{d}-\frac {3 a \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 a x}{2}}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}\) |
-((Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))) - ((-3*a*x)/2 + (2*a*Sin[c + d*x])/d - (3*a*Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^2
3.1.45.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
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 = 0.76 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {6 d x +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-7+\cos \left (2 d x +2 c \right )-2 \cos \left (d x +c \right )\right )}{4 a d}\) | \(43\) |
derivativedivides | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(74\) |
default | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(74\) |
risch | \(\frac {3 x}{2 a}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {\sin \left (2 d x +2 c \right )}{4 a d}\) | \(83\) |
norman | \(\frac {\frac {3 x}{2 a}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {9 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(149\) |
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {3 \, d x \cos \left (d x + c\right ) + 3 \, d x + {\left (\cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
1/2*(3*d*x*cos(d*x + c) + 3*d*x + (cos(d*x + c)^2 - cos(d*x + c) - 4)*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. 325 vs. \(2 (65) = 130\).
Time = 0.82 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.28 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=\begin {cases} \frac {3 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} + \frac {6 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} + \frac {3 d x}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {2 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {10 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {4 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\left (c \right )}}{a \cos {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
Piecewise((3*d*x*tan(c/2 + d*x/2)**4/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*ta n(c/2 + d*x/2)**2 + 2*a*d) + 6*d*x*tan(c/2 + d*x/2)**2/(2*a*d*tan(c/2 + d* x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) + 3*d*x/(2*a*d*tan(c/2 + d*x/ 2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) - 2*tan(c/2 + d*x/2)**5/(2*a*d* tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) - 10*tan(c/2 + d* x/2)**3/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d) - 4*tan(c/2 + d*x/2)/(2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d), Ne(d, 0)), (x*cos(c)**3/(a*cos(c) + a), True))
Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.75 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
-((sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 )/(a + 2*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + sin(d*x + c) /(a*(cos(d*x + c) + 1)))/d
Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {3 \, {\left (d x + c\right )}}{a} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]
1/2*(3*(d*x + c)/a - 2*tan(1/2*d*x + 1/2*c)/a - 2*(3*tan(1/2*d*x + 1/2*c)^ 3 + tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a))/d
Time = 14.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (c+d\,x\right )}{2}+3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]