Integrand size = 21, antiderivative size = 89 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} (5 A+6 C) x+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d} \] Output:
1/16*(5*A+6*C)*x+1/16*(5*A+6*C)*cos(d*x+c)*sin(d*x+c)/d+1/24*(5*A+6*C)*cos (d*x+c)^3*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*sin(d*x+c)/d
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {60 A c+72 c C+60 A d x+72 C d x+(45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x))}{192 d} \] Input:
Integrate[Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2),x]
Output:
(60*A*c + 72*c*C + 60*A*d*x + 72*C*d*x + (45*A + 48*C)*Sin[2*(c + d*x)] + (9*A + 6*C)*Sin[4*(c + d*x)] + A*Sin[6*(c + d*x)])/(192*d)
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4533, 3042, 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.
\(\displaystyle \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
\(\Big \downarrow \) 4533 |
\(\displaystyle \frac {1}{6} (5 A+6 C) \int \cos ^4(c+d x)dx+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} (5 A+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
Input:
Int[Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + ((5*A + 6*C)*((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))/6
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_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Time = 0.66 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {\left (45 A +48 C \right ) \sin \left (2 d x +2 c \right )+\left (9 A +6 C \right ) \sin \left (4 d x +4 c \right )+A \sin \left (6 d x +6 c \right )+60 x d \left (A +\frac {6 C}{5}\right )}{192 d}\) | \(61\) |
risch | \(\frac {5 A x}{16}+\frac {3 C x}{8}+\frac {A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {15 A \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 d}\) | \(85\) |
derivativedivides | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(86\) |
default | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(86\) |
norman | \(\frac {\left (-\frac {5 A}{16}-\frac {3 C}{8}\right ) x +\left (-\frac {45 A}{16}-\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {5 A}{16}+\frac {3 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {45 A}{16}+\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-\frac {\left (11 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (11 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}+\frac {\left (15 A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}+\frac {\left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-\frac {5 \left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{24 d}-\frac {5 \left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(341\) |
Input:
int(cos(d*x+c)^6*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/192*((45*A+48*C)*sin(2*d*x+2*c)+(9*A+6*C)*sin(4*d*x+4*c)+A*sin(6*d*x+6*c )+60*x*d*(A+6/5*C))/d
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (5 \, A + 6 \, C\right )} d x + {\left (8 \, A \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \] Input:
integrate(cos(d*x+c)^6*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
Output:
1/48*(3*(5*A + 6*C)*d*x + (8*A*cos(d*x + c)^5 + 2*(5*A + 6*C)*cos(d*x + c) ^3 + 3*(5*A + 6*C)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*(A+C*sec(d*x+c)**2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} + \frac {3 \, {\left (5 \, A + 6 \, C\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (5 \, A + 6 \, C\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (11 \, A + 10 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \] Input:
integrate(cos(d*x+c)^6*(A+C*sec(d*x+c)^2),x, algorithm="maxima")
Output:
1/48*(3*(d*x + c)*(5*A + 6*C) + (3*(5*A + 6*C)*tan(d*x + c)^5 + 8*(5*A + 6 *C)*tan(d*x + c)^3 + 3*(11*A + 10*C)*tan(d*x + c))/(tan(d*x + c)^6 + 3*tan (d*x + c)^4 + 3*tan(d*x + c)^2 + 1))/d
Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} + \frac {15 \, A \tan \left (d x + c\right )^{5} + 18 \, C \tan \left (d x + c\right )^{5} + 40 \, A \tan \left (d x + c\right )^{3} + 48 \, C \tan \left (d x + c\right )^{3} + 33 \, A \tan \left (d x + c\right ) + 30 \, C \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3}}}{48 \, d} \] Input:
integrate(cos(d*x+c)^6*(A+C*sec(d*x+c)^2),x, algorithm="giac")
Output:
1/48*(3*(d*x + c)*(5*A + 6*C) + (15*A*tan(d*x + c)^5 + 18*C*tan(d*x + c)^5 + 40*A*tan(d*x + c)^3 + 48*C*tan(d*x + c)^3 + 33*A*tan(d*x + c) + 30*C*ta n(d*x + c))/(tan(d*x + c)^2 + 1)^3)/d
Time = 12.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=x\,\left (\frac {5\,A}{16}+\frac {3\,C}{8}\right )+\frac {\left (\frac {5\,A}{16}+\frac {3\,C}{8}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (\frac {5\,A}{6}+C\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {11\,A}{16}+\frac {5\,C}{8}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6+3\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \] Input:
int(cos(c + d*x)^6*(A + C/cos(c + d*x)^2),x)
Output:
x*((5*A)/16 + (3*C)/8) + (tan(c + d*x)*((11*A)/16 + (5*C)/8) + tan(c + d*x )^3*((5*A)/6 + C) + tan(c + d*x)^5*((5*A)/16 + (3*C)/8))/(d*(3*tan(c + d*x )^2 + 3*tan(c + d*x)^4 + tan(c + d*x)^6 + 1))
Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -26 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +33 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +30 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +15 a d x +18 c d x}{48 d} \] Input:
int(cos(d*x+c)^6*(A+C*sec(d*x+c)^2),x)
Output:
(8*cos(c + d*x)*sin(c + d*x)**5*a - 26*cos(c + d*x)*sin(c + d*x)**3*a - 12 *cos(c + d*x)*sin(c + d*x)**3*c + 33*cos(c + d*x)*sin(c + d*x)*a + 30*cos( c + d*x)*sin(c + d*x)*c + 15*a*d*x + 18*c*d*x)/(48*d)