Integrand size = 27, antiderivative size = 91 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {3 \sec ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d} \] Output:
1/3*sec(d*x+c)^3/a^2/d-3/5*sec(d*x+c)^5/a^2/d+2/7*sec(d*x+c)^7/a^2/d-2/5*t an(d*x+c)^5/a^2/d-2/7*tan(d*x+c)^7/a^2/d
Time = 1.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.38 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\sec ^3(c+d x) (672-182 \cos (c+d x)-736 \cos (2 (c+d x))-39 \cos (3 (c+d x))+192 \cos (4 (c+d x))+13 \cos (5 (c+d x))+448 \sin (c+d x)-104 \sin (2 (c+d x))-144 \sin (3 (c+d x))-52 \sin (4 (c+d x))+48 \sin (5 (c+d x)))}{6720 a^2 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
Output:
(Sec[c + d*x]^3*(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*a^2*d*(1 + Sin[c + d*x])^2)
Time = 0.54 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3354, 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.
| \(\displaystyle \int \frac {\tan ^3(c+d x) \sec (c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3}{\cos (c+d x)^4 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \sec ^5(c+d x) (a-a \sin (c+d x))^2 \tan ^3(c+d x)dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 (a-a \sin (c+d x))^2}{\cos (c+d x)^8}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \tan ^3(c+d x) \sec ^5(c+d x)-2 a^2 \tan ^4(c+d x) \sec ^4(c+d x)+a^2 \tan ^5(c+d x) \sec ^3(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 a^2 \tan ^7(c+d x)}{7 d}-\frac {2 a^2 \tan ^5(c+d x)}{5 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}-\frac {3 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \sec ^3(c+d x)}{3 d}}{a^4}\) |
Input:
Int[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
Output:
((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))/a^4
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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Result contains complex when optimal does not.
Time = 1.63 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(\frac {\frac {8 i {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {8 \,{\mathrm e}^{7 i \left (d x +c \right )}}{3}+\frac {8 i {\mathrm e}^{4 i \left (d x +c \right )}}{15}-\frac {16 \,{\mathrm e}^{5 i \left (d x +c \right )}}{5}+\frac {24 i {\mathrm e}^{2 i \left (d x +c \right )}}{35}+\frac {88 \,{\mathrm e}^{3 i \left (d x +c \right )}}{105}-\frac {8 i}{35}-\frac {32 \,{\mathrm e}^{i \left (d x +c \right )}}{35}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{2}}\) | \(120\) |
| derivativedivides | \(\frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {14}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{d \,a^{2}}\) | \(130\) |
| default | \(\frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {14}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{d \,a^{2}}\) | \(130\) |
Input:
int(sec(d*x+c)*tan(d*x+c)^3/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
8/105*(35*I*exp(6*I*(d*x+c))+35*exp(7*I*(d*x+c))+7*I*exp(4*I*(d*x+c))-42*e xp(5*I*(d*x+c))+9*I*exp(2*I*(d*x+c))+11*exp(3*I*(d*x+c))-3*I-12*exp(I*(d*x +c)))/(exp(I*(d*x+c))+I)^7/(exp(I*(d*x+c))-I)^3/d/a^2
Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {24 \, \cos \left (d x + c\right )^{4} - 47 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + 5\right )} \sin \left (d x + c\right ) + 25}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
-1/105*(24*cos(d*x + c)^4 - 47*cos(d*x + c)^2 + 2*(6*cos(d*x + c)^4 - 9*co s(d*x + c)^2 + 5)*sin(d*x + c) + 25)/(a^2*d*cos(d*x + c)^5 - 2*a^2*d*cos(d *x + c)^3*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^3)
\[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)**3/(a+a*sin(d*x+c))**2,x)
Output:
Integral(tan(c + d*x)**3*sec(c + d*x)/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (81) = 162\).
Time = 0.04 (sec) , antiderivative size = 336, normalized size of antiderivative = 3.69 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \, {\left (\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}}{105 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
4/105*(4*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 91*sin(d*x + c)^4/(cos(d* x + c) + 1)^4 + 84*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 105*sin(d*x + c)^ 6/(cos(d*x + c) + 1)^6 + 1)/((a^2 + 4*a^2*sin(d*x + c)/(cos(d*x + c) + 1) + 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8*a^2*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 - 14*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 14*a^2*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 + 8*a^2*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 4*a^2*sin(d*x + c)^9/(cos(d* x + c) + 1)^9 - a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*d)
Time = 0.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.32 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {35 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1015 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1302 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 469 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 67}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
-1/840*(35*(3*tan(1/2*d*x + 1/2*c) - 1)/(a^2*(tan(1/2*d*x + 1/2*c) - 1)^3) - (105*tan(1/2*d*x + 1/2*c)^5 + 1015*tan(1/2*d*x + 1/2*c)^4 + 1330*tan(1/ 2*d*x + 1/2*c)^3 + 1302*tan(1/2*d*x + 1/2*c)^2 + 469*tan(1/2*d*x + 1/2*c) + 67)/(a^2*(tan(1/2*d*x + 1/2*c) + 1)^7))/d
Time = 35.05 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.27 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{105}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}+\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {52\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{a^2\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \] Input:
int(tan(c + d*x)^3/(cos(c + d*x)*(a + a*sin(c + d*x))^2),x)
Output:
((4*cos(c/2 + (d*x)/2)^10)/105 + (16*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/ 2))/105 + 4*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^6 + (16*cos(c/2 + (d*x )/2)^5*sin(c/2 + (d*x)/2)^5)/5 + (52*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/ 2)^4)/15 - (32*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^3)/105 + (4*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^2)/35)/(a^2*d*(cos(c/2 + (d*x)/2) - sin(c /2 + (d*x)/2))^3*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))^7)
Time = 0.20 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.71 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {35 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{2}-70 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right )^{4}+70 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}-140 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right )^{3}-70 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right ) \tan \left (d x +c \right )^{2}+140 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right )-35 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}+70 \cos \left (d x +c \right ) \sec \left (d x +c \right )+72 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right )-72 \cos \left (d x +c \right )-12 \sin \left (d x +c \right )^{5}+81 \sin \left (d x +c \right )^{4}+216 \sin \left (d x +c \right )^{3}+36 \sin \left (d x +c \right )^{2}-144 \sin \left (d x +c \right )-72}{105 \cos \left (d x +c \right ) a^{2} d \left (\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )-1\right )} \] Input:
int(sec(d*x+c)*tan(d*x+c)^3/(a+a*sin(d*x+c))^2,x)
Output:
(35*cos(c + d*x)*sec(c + d*x)*sin(c + d*x)**4*tan(c + d*x)**2 - 70*cos(c + d*x)*sec(c + d*x)*sin(c + d*x)**4 + 70*cos(c + d*x)*sec(c + d*x)*sin(c + d*x)**3*tan(c + d*x)**2 - 140*cos(c + d*x)*sec(c + d*x)*sin(c + d*x)**3 - 70*cos(c + d*x)*sec(c + d*x)*sin(c + d*x)*tan(c + d*x)**2 + 140*cos(c + d* x)*sec(c + d*x)*sin(c + d*x) - 35*cos(c + d*x)*sec(c + d*x)*tan(c + d*x)** 2 + 70*cos(c + d*x)*sec(c + d*x) + 72*cos(c + d*x)*sin(c + d*x)**4 + 144*c os(c + d*x)*sin(c + d*x)**3 - 144*cos(c + d*x)*sin(c + d*x) - 72*cos(c + d *x) - 12*sin(c + d*x)**5 + 81*sin(c + d*x)**4 + 216*sin(c + d*x)**3 + 36*s in(c + d*x)**2 - 144*sin(c + d*x) - 72)/(105*cos(c + d*x)*a**2*d*(sin(c + d*x)**4 + 2*sin(c + d*x)**3 - 2*sin(c + d*x) - 1))