Integrand size = 29, antiderivative size = 91 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {a^2 \sec ^3(c+d x)}{3 d}-\frac {3 a^2 \sec ^5(c+d x)}{5 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}+\frac {2 a^2 \tan ^5(c+d x)}{5 d}+\frac {2 a^2 \tan ^7(c+d x)}{7 d} \]
1/3*a^2*sec(d*x+c)^3/d-3/5*a^2*sec(d*x+c)^5/d+2/7*a^2*sec(d*x+c)^7/d+2/5*a ^2*tan(d*x+c)^5/d+2/7*a^2*tan(d*x+c)^7/d
Time = 4.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.53 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=-\frac {a^2 \sec ^7(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (-672+182 \cos (c+d x)+736 \cos (2 (c+d x))+39 \cos (3 (c+d x))-192 \cos (4 (c+d x))-13 \cos (5 (c+d x))+448 \sin (c+d x)-104 \sin (2 (c+d x))-144 \sin (3 (c+d x))-52 \sin (4 (c+d x))+48 \sin (5 (c+d x)))}{6720 d} \]
-1/6720*(a^2*Sec[c + d*x]^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-672 + 182*Cos[c + d*x] + 736*Cos[2*(c + d*x)] + 39*Cos[3*(c + d*x)] - 192*Cos[ 4*(c + d*x)] - 13*Cos[5*(c + d*x)] + 448*Sin[c + d*x] - 104*Sin[2*(c + d*x )] - 144*Sin[3*(c + d*x)] - 52*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)]))/d
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.
| \(\displaystyle \int \tan ^3(c+d x) \sec ^5(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 (a \sin (c+d x)+a)^2}{\cos (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \tan ^3(c+d x) \sec ^5(c+d x)+2 a^2 \tan ^4(c+d x) \sec ^4(c+d x)+a^2 \tan ^5(c+d x) \sec ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 \tan ^7(c+d x)}{7 d}+\frac {2 a^2 \tan ^5(c+d x)}{5 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}-\frac {3 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \sec ^3(c+d x)}{3 d}\) |
(a^2*Sec[c + d*x]^3)/(3*d) - (3*a^2*Sec[c + d*x]^5)/(5*d) + (2*a^2*Sec[c + d*x]^7)/(7*d) + (2*a^2*Tan[c + d*x]^5)/(5*d) + (2*a^2*Tan[c + d*x]^7)/(7* d)
3.9.94.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.24
| method | result | size |
| parallelrisch | \(-\frac {4 a^{2} \left (105 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-84 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+91 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{105 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(113\) |
| risch | \(\frac {8 a^{2} \left (-35 i {\mathrm e}^{6 i \left (d x +c \right )}+35 \,{\mathrm e}^{7 i \left (d x +c \right )}-7 i {\mathrm e}^{4 i \left (d x +c \right )}-42 \,{\mathrm e}^{5 i \left (d x +c \right )}-9 i {\mathrm e}^{2 i \left (d x +c \right )}+11 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i-12 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{105 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{7} d}\) | \(120\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{35}\right )+2 a^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d}\) | \(248\) |
| default | \(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{35}\right )+2 a^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d}\) | \(248\) |
-4/105*a^2*(105*tan(1/2*d*x+1/2*c)^6-84*tan(1/2*d*x+1/2*c)^5+91*tan(1/2*d* x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2-4*tan(1/2*d*x+1/2 *c)+1)/d/(tan(1/2*d*x+1/2*c)+1)^3/(tan(1/2*d*x+1/2*c)-1)^7
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=-\frac {24 \, a^{2} \cos \left (d x + c\right )^{4} - 47 \, a^{2} \cos \left (d x + c\right )^{2} + 25 \, a^{2} - 2 \, {\left (6 \, a^{2} \cos \left (d x + c\right )^{4} - 9 \, a^{2} \cos \left (d x + c\right )^{2} + 5 \, a^{2}\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{3}\right )}} \]
-1/105*(24*a^2*cos(d*x + c)^4 - 47*a^2*cos(d*x + c)^2 + 25*a^2 - 2*(6*a^2* cos(d*x + c)^4 - 9*a^2*cos(d*x + c)^2 + 5*a^2)*sin(d*x + c))/(d*cos(d*x + c)^5 + 2*d*cos(d*x + c)^3*sin(d*x + c) - 2*d*cos(d*x + c)^3)
Timed out. \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {6 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} a^{2} + \frac {{\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} a^{2}}{\cos \left (d x + c\right )^{7}} - \frac {3 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{2}}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]
1/105*(6*(5*tan(d*x + c)^7 + 7*tan(d*x + c)^5)*a^2 + (35*cos(d*x + c)^4 - 42*cos(d*x + c)^2 + 15)*a^2/cos(d*x + c)^7 - 3*(7*cos(d*x + c)^2 - 5)*a^2/ cos(d*x + c)^7)/d
Time = 0.37 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.52 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=-\frac {\frac {35 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} - \frac {105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1015 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1302 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 469 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 67 \, a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{7}}}{840 \, d} \]
-1/840*(35*(3*a^2*tan(1/2*d*x + 1/2*c) + a^2)/(tan(1/2*d*x + 1/2*c) + 1)^3 - (105*a^2*tan(1/2*d*x + 1/2*c)^5 - 1015*a^2*tan(1/2*d*x + 1/2*c)^4 + 133 0*a^2*tan(1/2*d*x + 1/2*c)^3 - 1302*a^2*tan(1/2*d*x + 1/2*c)^2 + 469*a^2*t an(1/2*d*x + 1/2*c) - 67*a^2)/(tan(1/2*d*x + 1/2*c) - 1)^7)/d
Time = 15.17 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.36 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {4\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-84\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+105\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{105\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3} \]
(4*a^2*cos(c/2 + (d*x)/2)^3*(cos(c/2 + (d*x)/2)^7 + 105*cos(c/2 + (d*x)/2) *sin(c/2 + (d*x)/2)^6 - 4*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) - 84*cos (c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^5 + 91*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^4 + 8*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^3 + 3*cos(c/2 + (d *x)/2)^5*sin(c/2 + (d*x)/2)^2))/(105*d*(cos(c/2 + (d*x)/2) - sin(c/2 + (d* x)/2))^7*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))^3)