Integrand size = 21, antiderivative size = 99 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{16 a}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}+\frac {\sin ^5(c+d x)}{5 a d} \]
-1/16*x/a-1/16*cos(d*x+c)*sin(d*x+c)/a/d+1/8*cos(d*x+c)^3*sin(d*x+c)/a/d+1 /6*cos(d*x+c)^3*sin(d*x+c)^3/a/d+1/5*sin(d*x+c)^5/a/d
Time = 0.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (75 c-60 d x+120 \sin (c+d x)+15 \sin (2 (c+d x))-60 \sin (3 (c+d x))+15 \sin (4 (c+d x))+12 \sin (5 (c+d x))-5 \sin (6 (c+d x))-75 \tan \left (\frac {c}{2}\right )\right )}{480 a d (1+\sec (c+d x))} \]
(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(75*c - 60*d*x + 120*Sin[c + d*x] + 15*Si n[2*(c + d*x)] - 60*Sin[3*(c + d*x)] + 15*Sin[4*(c + d*x)] + 12*Sin[5*(c + d*x)] - 5*Sin[6*(c + d*x)] - 75*Tan[c/2]))/(480*a*d*(1 + Sec[c + d*x]))
Time = 0.64 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4360, 25, 25, 3042, 3318, 3042, 3044, 15, 3048, 3042, 3048, 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.
| \(\displaystyle \int \frac {\sin ^6(c+d x)}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^6}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^6(c+d x) \cos (c+d x)}{a (-\cos (c+d x))-a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \sin ^6(c+d x)}{\cos (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^6(c+d x) \cos (c+d x)}{a \cos (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^6}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos (c+d x) \sin ^4(c+d x)dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x) \sin (c+d x)^4dx}{a}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\int \sin ^4(c+d x)d\sin (c+d x)}{a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\frac {1}{2} \int \cos ^2(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\frac {1}{2} \int \cos (c+d x)^2 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}}{a}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\frac {1}{2} \left (\frac {1}{4} \int \cos ^2(c+d x)dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\frac {1}{2} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sin ^5(c+d x)}{5 a d}-\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}}{a}\) |
Sin[c + d*x]^5/(5*a*d) - (-1/6*(Cos[c + d*x]^3*Sin[c + d*x]^3)/d + (-1/4*( Cos[c + d*x]^3*Sin[c + d*x])/d + (x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) /4)/2)/a
3.1.66.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.59 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78
| method | result | size |
| parallelrisch | \(\frac {-60 d x +120 \sin \left (d x +c \right )+12 \sin \left (5 d x +5 c \right )-60 \sin \left (3 d x +3 c \right )-5 \sin \left (6 d x +6 c \right )+15 \sin \left (4 d x +4 c \right )+15 \sin \left (2 d x +2 c \right )}{960 d a}\) | \(77\) |
| risch | \(-\frac {x}{16 a}+\frac {\sin \left (d x +c \right )}{8 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}+\frac {\sin \left (5 d x +5 c \right )}{80 d a}+\frac {\sin \left (4 d x +4 c \right )}{64 d a}-\frac {\sin \left (3 d x +3 c \right )}{16 d a}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(107\) |
| derivativedivides | \(\frac {-\frac {64 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{512}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1536}-\frac {223 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1280}-\frac {33 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1280}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1536}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(116\) |
| default | \(\frac {-\frac {64 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{512}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1536}-\frac {223 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1280}-\frac {33 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1280}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1536}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(116\) |
| norman | \(\frac {-\frac {x}{16 a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}+\frac {33 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {223 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 a d}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 a d}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(238\) |
1/960*(-60*d*x+120*sin(d*x+c)+12*sin(5*d*x+5*c)-60*sin(3*d*x+3*c)-5*sin(6* d*x+6*c)+15*sin(4*d*x+4*c)+15*sin(2*d*x+2*c))/d/a
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {15 \, d x + {\left (40 \, \cos \left (d x + c\right )^{5} - 48 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{3} + 96 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) - 48\right )} \sin \left (d x + c\right )}{240 \, a d} \]
-1/240*(15*d*x + (40*cos(d*x + c)^5 - 48*cos(d*x + c)^4 - 70*cos(d*x + c)^ 3 + 96*cos(d*x + c)^2 + 15*cos(d*x + c) - 48)*sin(d*x + c))/(a*d)
\[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (89) = 178\).
Time = 0.30 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.81 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {85 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {198 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1338 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {85 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a + \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \]
1/120*((15*sin(d*x + c)/(cos(d*x + c) + 1) + 85*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 198*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 1338*sin(d*x + c)^7/ (cos(d*x + c) + 1)^7 - 85*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 15*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a + 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1) ^2 + 15*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a*sin(d*x + c)^6/(cos(d *x + c) + 1)^6 + 15*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 1 5*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {15 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 85 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1338 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 198 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 85 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \]
-1/240*(15*(d*x + c)/a + 2*(15*tan(1/2*d*x + 1/2*c)^11 + 85*tan(1/2*d*x + 1/2*c)^9 - 1338*tan(1/2*d*x + 1/2*c)^7 - 198*tan(1/2*d*x + 1/2*c)^5 - 85*t an(1/2*d*x + 1/2*c)^3 - 15*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a))/d
Time = 16.46 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {223\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}-\frac {x}{16\,a} \]