Integrand size = 19, antiderivative size = 76 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 a x}{8}+\frac {b \sin (c+d x)}{d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {b \sin ^3(c+d x)}{3 d} \]
3/8*a*x+b*sin(d*x+c)/d+3/8*a*cos(d*x+c)*sin(d*x+c)/d+1/4*a*cos(d*x+c)^3*si n(d*x+c)/d-1/3*b*sin(d*x+c)^3/d
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 a (c+d x)}{8 d}+\frac {b \sin (c+d x)}{d}-\frac {b \sin ^3(c+d x)}{3 d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \sin (4 (c+d x))}{32 d} \]
(3*a*(c + d*x))/(8*d) + (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) + (a *Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)
Time = 0.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 4274, 3042, 3113, 2009, 3115, 3042, 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.
\(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle a \int \cos ^4(c+d x)dx+b \int \cos ^3(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+b \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {b \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {b \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {b \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {b \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {b \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {b \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
-((b*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d) + a*((Cos[c + d*x]^3*Sin[c + d *x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4)
3.5.54.3.1 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]
Time = 0.89 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {36 a x d +72 \sin \left (d x +c \right ) b +3 a \sin \left (4 d x +4 c \right )+8 b \sin \left (3 d x +3 c \right )+24 a \sin \left (2 d x +2 c \right )}{96 d}\) | \(57\) |
derivativedivides | \(\frac {a \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(60\) |
default | \(\frac {a \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(60\) |
risch | \(\frac {3 a x}{8}+\frac {3 b \sin \left (d x +c \right )}{4 d}+\frac {a \sin \left (4 d x +4 c \right )}{32 d}+\frac {b \sin \left (3 d x +3 c \right )}{12 d}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(63\) |
norman | \(\frac {\frac {3 a x}{8}+\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {9 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}-\frac {\left (5 a -8 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (5 a +8 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (9 a -40 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {\left (9 a +40 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(172\) |
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\frac {9 \, a d x + {\left (6 \, a \cos \left (d x + c\right )^{3} + 8 \, b \cos \left (d x + c\right )^{2} + 9 \, a \cos \left (d x + c\right ) + 16 \, b\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(9*a*d*x + (6*a*cos(d*x + c)^3 + 8*b*cos(d*x + c)^2 + 9*a*cos(d*x + c ) + 16*b)*sin(d*x + c))/d
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b}{96 \, d} \]
1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a - 32*(si n(d*x + c)^3 - 3*sin(d*x + c))*b)/d
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (68) = 136\).
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.84 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\frac {9 \, {\left (d x + c\right )} a - \frac {2 \, {\left (15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
1/24*(9*(d*x + c)*a - 2*(15*a*tan(1/2*d*x + 1/2*c)^7 - 24*b*tan(1/2*d*x + 1/2*c)^7 - 9*a*tan(1/2*d*x + 1/2*c)^5 - 40*b*tan(1/2*d*x + 1/2*c)^5 + 9*a* tan(1/2*d*x + 1/2*c)^3 - 40*b*tan(1/2*d*x + 1/2*c)^3 - 15*a*tan(1/2*d*x + 1/2*c) - 24*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 14.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3\,a\,x}{8}+\frac {2\,b\,\sin \left (c+d\,x\right )}{3\,d}+\frac {3\,a\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]