Integrand size = 19, antiderivative size = 74 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \cos ^7(c+d x)}{7 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d} \] Output:
-1/7*b*cos(d*x+c)^7/d+a*sin(d*x+c)/d-a*sin(d*x+c)^3/d+3/5*a*sin(d*x+c)^5/d -1/7*a*sin(d*x+c)^7/d
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \cos ^7(c+d x)}{7 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + b*Tan[c + d*x]),x]
Output:
-1/7*(b*Cos[c + d*x]^7)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3 *a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 3967, 3042, 3113, 2009}
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.
\(\displaystyle \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \tan (c+d x)}{\sec (c+d x)^7}dx\) |
\(\Big \downarrow \) 3967 |
\(\displaystyle a \int \cos ^7(c+d x)dx-\frac {b \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^7dx-\frac {b \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -\frac {a \int \left (-\sin ^6(c+d x)+3 \sin ^4(c+d x)-3 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{d}-\frac {b \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}-\frac {b \cos ^7(c+d x)}{7 d}\) |
Input:
Int[Cos[c + d*x]^7*(a + b*Tan[c + d*x]),x]
Output:
-1/7*(b*Cos[c + d*x]^7)/d - (a*(-Sin[c + d*x] + Sin[c + d*x]^3 - (3*Sin[c + d*x]^5)/5 + Sin[c + d*x]^7/7))/d
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[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Time = 36.89 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {-\frac {b \cos \left (d x +c \right )^{7}}{7}+\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(56\) |
default | \(\frac {-\frac {b \cos \left (d x +c \right )^{7}}{7}+\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(56\) |
risch | \(-\frac {5 b \cos \left (d x +c \right )}{64 d}+\frac {35 a \sin \left (d x +c \right )}{64 d}-\frac {b \cos \left (7 d x +7 c \right )}{448 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}-\frac {b \cos \left (5 d x +5 c \right )}{64 d}+\frac {7 a \sin \left (5 d x +5 c \right )}{320 d}-\frac {3 b \cos \left (3 d x +3 c \right )}{64 d}+\frac {7 a \sin \left (3 d x +3 c \right )}{64 d}\) | \(116\) |
Input:
int(cos(d*x+c)^7*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/7*b*cos(d*x+c)^7+1/7*a*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos (d*x+c)^2)*sin(d*x+c))
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {5 \, b \cos \left (d x + c\right )^{7} - {\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{35 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
-1/35*(5*b*cos(d*x + c)^7 - (5*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 + 8*a *cos(d*x + c)^2 + 16*a)*sin(d*x + c))/d
\[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \cos ^{7}{\left (c + d x \right )}\, dx \] Input:
integrate(cos(d*x+c)**7*(a+b*tan(d*x+c)),x)
Output:
Integral((a + b*tan(c + d*x))*cos(c + d*x)**7, x)
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {5 \, b \cos \left (d x + c\right )^{7} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a}{35 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
-1/35*(5*b*cos(d*x + c)^7 + (5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin (d*x + c)^3 - 35*sin(d*x + c))*a)/d
Leaf count of result is larger than twice the leaf count of optimal. 49370 vs. \(2 (68) = 136\).
Time = 10.87 (sec) , antiderivative size = 49370, normalized size of antiderivative = 667.16 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
1/17920*(315*pi*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1 /2*c) + tan(1/2*d*x)^2 - tan(1/2*c)^2 + 2*tan(1/2*c) - 1)*sgn(tan(1/2*d*x) ^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)*tan(1/2*c)^2 - tan(1/2*d*x)^2 + tan(1/2*c )^2 + 2*tan(1/2*d*x) - 1)*tan(1/2*d*x)^14*tan(1/2*c)^14 + 315*pi*b*sgn(tan (1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) + tan(1/2*d*x)^2 - tan(1/2*c)^2 - 2*tan(1/2*c) - 1)*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1 /2*d*x)*tan(1/2*c)^2 - tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 1) *tan(1/2*d*x)^14*tan(1/2*c)^14 - 735*pi*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)*tan(1/2*c)^2 - tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2* d*x) - 1)*tan(1/2*d*x)^14*tan(1/2*c)^14 + 735*pi*b*sgn(tan(1/2*d*x)^2*tan( 1/2*c)^2 - 2*tan(1/2*d*x)*tan(1/2*c)^2 - tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2 *tan(1/2*d*x) - 1)*tan(1/2*d*x)^14*tan(1/2*c)^14 - 630*pi*b*sgn(tan(1/2*d* x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c )^2 + 1)*tan(1/2*d*x)^14*tan(1/2*c)^14 + 2205*pi*b*sgn(tan(1/2*d*x)^2*tan( 1/2*c)^2 + 2*tan(1/2*d*x)^2*tan(1/2*c) + tan(1/2*d*x)^2 - tan(1/2*c)^2 + 2 *tan(1/2*c) - 1)*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)*tan(1/2* c)^2 - tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) - 1)*tan(1/2*d*x)^14 *tan(1/2*c)^12 + 2205*pi*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x )^2*tan(1/2*c) + tan(1/2*d*x)^2 - tan(1/2*c)^2 - 2*tan(1/2*c) - 1)*sgn(tan (1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)*tan(1/2*c)^2 - tan(1/2*d*x)^2...
Time = 0.80 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.18 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=\frac {16\,a\,\sin \left (c+d\,x\right )}{35\,d}-\frac {b\,{\cos \left (c+d\,x\right )}^7}{7\,d}+\frac {8\,a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d} \] Input:
int(cos(c + d*x)^7*(a + b*tan(c + d*x)),x)
Output:
(16*a*sin(c + d*x))/(35*d) - (b*cos(c + d*x)^7)/(7*d) + (8*a*cos(c + d*x)^ 2*sin(c + d*x))/(35*d) + (6*a*cos(c + d*x)^4*sin(c + d*x))/(35*d) + (a*cos (c + d*x)^6*sin(c + d*x))/(7*d)
Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.50 \[ \int \cos ^7(c+d x) (a+b \tan (c+d x)) \, dx=\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b -15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b +15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b -5 \cos \left (d x +c \right ) b -5 \sin \left (d x +c \right )^{7} a +21 \sin \left (d x +c \right )^{5} a -35 \sin \left (d x +c \right )^{3} a +35 \sin \left (d x +c \right ) a +5 b}{35 d} \] Input:
int(cos(d*x+c)^7*(a+b*tan(d*x+c)),x)
Output:
(5*cos(c + d*x)*sin(c + d*x)**6*b - 15*cos(c + d*x)*sin(c + d*x)**4*b + 15 *cos(c + d*x)*sin(c + d*x)**2*b - 5*cos(c + d*x)*b - 5*sin(c + d*x)**7*a + 21*sin(c + d*x)**5*a - 35*sin(c + d*x)**3*a + 35*sin(c + d*x)*a + 5*b)/(3 5*d)