Integrand size = 27, antiderivative size = 101 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {11 a^3 x}{2}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {19 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \] Output:
11/2*a^3*x-3*a^3*cos(d*x+c)/d+2/3*a^3*cos(d*x+c)/d/(1-sin(d*x+c))^2-19/3*a ^3*cos(d*x+c)/d/(1-sin(d*x+c))-1/2*a^3*cos(d*x+c)*sin(d*x+c)/d
Time = 7.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.57 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=-\frac {a^3 \left (-3 (89+132 c+132 d x) \cos \left (\frac {1}{2} (c+d x)\right )+(403+132 c+132 d x) \cos \left (\frac {3}{2} (c+d x)\right )+3 \left (-9 \cos \left (\frac {5}{2} (c+d x)\right )+\cos \left (\frac {7}{2} (c+d x)\right )+2 (86+88 c+88 d x+(-43+44 c+44 d x) \cos (c+d x)-10 \cos (2 (c+d x))-\cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \] Input:
Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]
Output:
-1/48*(a^3*(-3*(89 + 132*c + 132*d*x)*Cos[(c + d*x)/2] + (403 + 132*c + 13 2*d*x)*Cos[(3*(c + d*x))/2] + 3*(-9*Cos[(5*(c + d*x))/2] + Cos[(7*(c + d*x ))/2] + 2*(86 + 88*c + 88*d*x + (-43 + 44*c + 44*d*x)*Cos[c + d*x] - 10*Co s[2*(c + d*x)] - Cos[3*(c + d*x)])*Sin[(c + d*x)/2])))/(d*(Cos[(c + d*x)/2 ] - Sin[(c + d*x)/2])^3)
Time = 0.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3351, 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.
| \(\displaystyle \int \tan ^3(c+d x) \sec (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 (a \sin (c+d x)+a)^3}{\cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle a^4 \int \left (\frac {\sin ^2(c+d x)}{a}+\frac {3 \sin (c+d x)}{a}+\frac {5}{a}-\frac {7}{a (1-\sin (c+d x))}+\frac {2}{a (1-\sin (c+d x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (-\frac {3 \cos (c+d x)}{a d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a d}-\frac {19 \cos (c+d x)}{3 a d (1-\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (1-\sin (c+d x))^2}+\frac {11 x}{2 a}\right )\) |
Input:
Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]
Output:
a^4*((11*x)/(2*a) - (3*Cos[c + d*x])/(a*d) + (2*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])^2) - (19*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Result contains complex when optimal does not.
Time = 5.72 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {11 a^{3} x}{2}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 a^{3} \left (-36 i {\mathrm e}^{i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}-19\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3}}\) | \(125\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(246\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(246\) |
Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
11/2*a^3*x+1/8*I*a^3/d*exp(2*I*(d*x+c))-3/2*a^3/d*exp(I*(d*x+c))-3/2*a^3/d *exp(-I*(d*x+c))-1/8*I*a^3/d*exp(-2*I*(d*x+c))-2/3*a^3*(-36*I*exp(I*(d*x+c ))+21*exp(2*I*(d*x+c))-19)/d/(exp(I*(d*x+c))-I)^3
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (89) = 178\).
Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.94 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {3 \, a^{3} \cos \left (d x + c\right )^{4} - 12 \, a^{3} \cos \left (d x + c\right )^{3} - 66 \, a^{3} d x - 4 \, a^{3} + {\left (33 \, a^{3} d x + 53 \, a^{3}\right )} \cos \left (d x + c\right )^{2} - {\left (33 \, a^{3} d x - 64 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{3} \cos \left (d x + c\right )^{3} - 66 \, a^{3} d x + 15 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} - {\left (33 \, a^{3} d x - 68 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^3,x, algorithm="fricas" )
Output:
1/6*(3*a^3*cos(d*x + c)^4 - 12*a^3*cos(d*x + c)^3 - 66*a^3*d*x - 4*a^3 + ( 33*a^3*d*x + 53*a^3)*cos(d*x + c)^2 - (33*a^3*d*x - 64*a^3)*cos(d*x + c) - (3*a^3*cos(d*x + c)^3 - 66*a^3*d*x + 15*a^3*cos(d*x + c)^2 + 4*a^3 - (33* a^3*d*x - 68*a^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d* x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)
\[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=a^{3} \left (\int \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))**3*tan(d*x+c)**3,x)
Output:
a**3*(Integral(tan(c + d*x)**3*sec(c + d*x), x) + Integral(3*sin(c + d*x)* tan(c + d*x)**3*sec(c + d*x), x) + Integral(3*sin(c + d*x)**2*tan(c + d*x) **3*sec(c + d*x), x) + Integral(sin(c + d*x)**3*tan(c + d*x)**3*sec(c + d* x), x))
Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.44 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {{\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{3} + 6 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^3,x, algorithm="maxima" )
Output:
1/6*((2*tan(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*tan(d*x + c))*a^3 + 6*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^3 - 6*a^3*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d*x + c)) - 2*(3*cos(d*x + c)^2 - 1)*a^3/cos(d*x + c)^3)/d
Time = 0.59 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {33 \, {\left (d x + c\right )} a^{3} + \frac {6 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {4 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^3,x, algorithm="giac")
Output:
1/6*(33*(d*x + c)*a^3 + 6*(a^3*tan(1/2*d*x + 1/2*c)^3 - 6*a^3*tan(1/2*d*x + 1/2*c)^2 - a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1 )^2 + 4*(15*a^3*tan(1/2*d*x + 1/2*c)^2 - 36*a^3*tan(1/2*d*x + 1/2*c) + 17* a^3)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d
Time = 34.96 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {11\,a^3\,x}{2}+\frac {\frac {11\,a^3\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {33\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (99\,c+99\,d\,x-246\right )}{6}\right )-\frac {a^3\,\left (33\,c+33\,d\,x-104\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {33\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (99\,c+99\,d\,x-66\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {55\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (165\,c+165\,d\,x-198\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {55\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (165\,c+165\,d\,x-322\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {77\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (231\,c+231\,d\,x-308\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {77\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (231\,c+231\,d\,x-420\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \] Input:
int((tan(c + d*x)^3*(a + a*sin(c + d*x))^3)/cos(c + d*x),x)
Output:
(11*a^3*x)/2 + ((11*a^3*(c + d*x))/2 - tan(c/2 + (d*x)/2)*((33*a^3*(c + d* x))/2 - (a^3*(99*c + 99*d*x - 246))/6) - (a^3*(33*c + 33*d*x - 104))/6 + t an(c/2 + (d*x)/2)^6*((33*a^3*(c + d*x))/2 - (a^3*(99*c + 99*d*x - 66))/6) - tan(c/2 + (d*x)/2)^5*((55*a^3*(c + d*x))/2 - (a^3*(165*c + 165*d*x - 198 ))/6) + tan(c/2 + (d*x)/2)^2*((55*a^3*(c + d*x))/2 - (a^3*(165*c + 165*d*x - 322))/6) + tan(c/2 + (d*x)/2)^4*((77*a^3*(c + d*x))/2 - (a^3*(231*c + 2 31*d*x - 308))/6) - tan(c/2 + (d*x)/2)^3*((77*a^3*(c + d*x))/2 - (a^3*(231 *c + 231*d*x - 420))/6))/(d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c/2 + (d*x)/2) ^2 + 1)^2)
Time = 0.20 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.93 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {a^{3} \left (-3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+33 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d x +41 \cos \left (d x +c \right ) \sin \left (d x +c \right )-33 \cos \left (d x +c \right ) d x -22 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )^{4}+15 \sin \left (d x +c \right )^{3}+33 \sin \left (d x +c \right )^{2} d x -89 \sin \left (d x +c \right )^{2}-66 \sin \left (d x +c \right ) d x +41 \sin \left (d x +c \right )+33 d x +22\right )}{6 d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )-\cos \left (d x +c \right )+\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1\right )} \] Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^3,x)
Output:
(a**3*( - 3*cos(c + d*x)*sin(c + d*x)**3 - 12*cos(c + d*x)*sin(c + d*x)**2 + 33*cos(c + d*x)*sin(c + d*x)*d*x + 41*cos(c + d*x)*sin(c + d*x) - 33*co s(c + d*x)*d*x - 22*cos(c + d*x) + 3*sin(c + d*x)**4 + 15*sin(c + d*x)**3 + 33*sin(c + d*x)**2*d*x - 89*sin(c + d*x)**2 - 66*sin(c + d*x)*d*x + 41*s in(c + d*x) + 33*d*x + 22))/(6*d*(cos(c + d*x)*sin(c + d*x) - cos(c + d*x) + sin(c + d*x)**2 - 2*sin(c + d*x) + 1))