Integrand size = 21, antiderivative size = 94 \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sin (c+d x)}{a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\cos ^2(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {4 \sin ^3(c+d x)}{3 a d} \]
-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)^2*sin(d *x+c)/d/(a+a*sec(d*x+c))-4/3*sin(d*x+c)^3/a/d
Time = 1.00 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-36 d x \cos \left (\frac {d x}{2}\right )-36 d x \cos \left (c+\frac {d x}{2}\right )+69 \sin \left (\frac {d x}{2}\right )+21 \sin \left (c+\frac {d x}{2}\right )+18 \sin \left (c+\frac {3 d x}{2}\right )+18 \sin \left (2 c+\frac {3 d x}{2}\right )-2 \sin \left (2 c+\frac {5 d x}{2}\right )-2 \sin \left (3 c+\frac {5 d x}{2}\right )+\sin \left (3 c+\frac {7 d x}{2}\right )+\sin \left (4 c+\frac {7 d x}{2}\right )\right )}{48 a d} \]
(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 = 0.47 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4306, 25, 3042, 4274, 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.
\(\displaystyle \int \frac {\cos ^3(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )}dx\) |
\(\Big \downarrow \) 4306 |
\(\displaystyle -\frac {\int -\cos ^3(c+d x) (4 a-3 a \sec (c+d x))dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cos ^3(c+d x) (4 a-3 a \sec (c+d x))dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {4 a-3 a \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {4 a \int \cos ^3(c+d x)dx-3 a \int \cos ^2(c+d x)dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx-3 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {-\frac {4 a \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}-3 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-3 a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {-3 a \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {4 a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {-\frac {4 a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}-3 a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\) |
-((Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))) + (-3*a*(x/2 + ( Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (4*a*(-Sin[c + d*x] + Sin[c + d*x]^3/3 ))/d)/a^2
3.1.50.3.1 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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x]))), x] - Simp[1/a^2 Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0 ]
Time = 0.45 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {\left (31+\cos \left (3 d x +3 c \right )-\cos \left (2 d x +2 c \right )+17 \cos \left (d x +c \right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-18 d x}{12 d a}\) | \(54\) |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(85\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(85\) |
risch | \(-\frac {3 x}{2 a}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )}}{8 d a}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )}}{8 d a}+\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {\sin \left (3 d x +3 c \right )}{12 d a}-\frac {\sin \left (2 d x +2 c \right )}{4 d a}\) | \(100\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}-\frac {3 x}{2 a}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}-\frac {9 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {9 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(148\) |
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {9 \, d x \cos \left (d x + c\right ) + 9 \, d x - {\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 7 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
-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)
Timed out. \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
Time = 0.40 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.87 \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{3 \, d} \]
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 = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {9 \, {\left (d x + c\right )}}{a} - \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \]
-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 = 12.87 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {3\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-\frac {\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{12}+\frac {\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{24}}{a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {3\,x}{2\,a} \]