Integrand size = 25, antiderivative size = 99 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}-\frac {3 \sec (c+d x)}{35 a d (a+a \sin (c+d x))^2}-\frac {3 \sec (c+d x)}{35 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {6 \tan (c+d x)}{35 a^3 d} \] Output:
1/7*sec(d*x+c)/d/(a+a*sin(d*x+c))^3-3/35*sec(d*x+c)/a/d/(a+a*sin(d*x+c))^2 -3/35*sec(d*x+c)/d/(a^3+a^3*sin(d*x+c))+6/35*tan(d*x+c)/a^3/d
Time = 1.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec (c+d x) (560+182 \cos (c+d x)-672 \cos (2 (c+d x))-78 \cos (3 (c+d x))+48 \cos (4 (c+d x))+672 \sin (c+d x)+182 \sin (2 (c+d x))-288 \sin (3 (c+d x))-13 \sin (4 (c+d x)))}{2240 a^3 d (1+\sin (c+d x))^3} \] Input:
Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
(Sec[c + d*x]*(560 + 182*Cos[c + d*x] - 672*Cos[2*(c + d*x)] - 78*Cos[3*(c + d*x)] + 48*Cos[4*(c + d*x)] + 672*Sin[c + d*x] + 182*Sin[2*(c + d*x)] - 288*Sin[3*(c + d*x)] - 13*Sin[4*(c + d*x)]))/(2240*a^3*d*(1 + Sin[c + d*x ])^3)
Time = 0.59 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3338, 3042, 3151, 3042, 3151, 3042, 4254, 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.
| \(\displaystyle \int \frac {\tan (c+d x) \sec (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)}{\cos (c+d x)^2 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle \frac {3 \int \frac {\sec ^2(c+d x)}{(\sin (c+d x) a+a)^2}dx}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {1}{\cos (c+d x)^2 (\sin (c+d x) a+a)^2}dx}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {3 \left (\frac {3 \int \frac {\sec ^2(c+d x)}{\sin (c+d x) a+a}dx}{5 a}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}\right )}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {3 \int \frac {1}{\cos (c+d x)^2 (\sin (c+d x) a+a)}dx}{5 a}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}\right )}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {3 \left (\frac {3 \left (\frac {2 \int \sec ^2(c+d x)dx}{3 a}-\frac {\sec (c+d x)}{3 d (a \sin (c+d x)+a)}\right )}{5 a}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}\right )}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {3 \left (\frac {2 \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{3 a}-\frac {\sec (c+d x)}{3 d (a \sin (c+d x)+a)}\right )}{5 a}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}\right )}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {3 \left (\frac {3 \left (-\frac {2 \int 1d(-\tan (c+d x))}{3 a d}-\frac {\sec (c+d x)}{3 d (a \sin (c+d x)+a)}\right )}{5 a}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}\right )}{7 a}+\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}+\frac {3 \left (\frac {3 \left (\frac {2 \tan (c+d x)}{3 a d}-\frac {\sec (c+d x)}{3 d (a \sin (c+d x)+a)}\right )}{5 a}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}\right )}{7 a}\) |
Input:
Int[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
Sec[c + d*x]/(7*d*(a + a*Sin[c + d*x])^3) + (3*(-1/5*Sec[c + d*x]/(d*(a + a*Sin[c + d*x])^2) + (3*(-1/3*Sec[c + d*x]/(d*(a + a*Sin[c + d*x])) + (2*T an[c + d*x])/(3*a*d)))/(5*a)))/(7*a)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 3.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(-\frac {4 i \left (-42 \,{\mathrm e}^{2 i \left (d x +c \right )}-18 i {\mathrm e}^{i \left (d x +c \right )}+3+35 \,{\mathrm e}^{4 i \left (d x +c \right )}+42 i {\mathrm e}^{3 i \left (d x +c \right )}\right )}{35 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} d \,a^{3}}\) | \(86\) |
| derivativedivides | \(\frac {-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {34}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {7}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {9}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {4}{32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32}}{a^{3} d}\) | \(130\) |
| default | \(\frac {-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {34}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {7}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {9}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {4}{32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32}}{a^{3} d}\) | \(130\) |
Input:
int(sec(d*x+c)*tan(d*x+c)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-4/35*I*(-42*exp(2*I*(d*x+c))-18*I*exp(I*(d*x+c))+3+35*exp(4*I*(d*x+c))+42 *I*exp(3*I*(d*x+c)))/(exp(I*(d*x+c))-I)/(exp(I*(d*x+c))+I)^7/d/a^3
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {6 \, \cos \left (d x + c\right )^{4} - 27 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (6 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 20}{35 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-1/35*(6*cos(d*x + c)^4 - 27*cos(d*x + c)^2 - 3*(6*cos(d*x + c)^2 - 5)*sin (d*x + c) + 20)/(3*a^3*d*cos(d*x + c)^3 - 4*a^3*d*cos(d*x + c) + (a^3*d*co s(d*x + c)^3 - 4*a^3*d*cos(d*x + c))*sin(d*x + c))
\[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\tan {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+a*sin(d*x+c))**3,x)
Output:
Integral(tan(c + d*x)*sec(c + d*x)/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (91) = 182\).
Time = 0.04 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.93 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {56 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {70 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}}{35 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-2/35*(6*sin(d*x + c)/(cos(d*x + c) + 1) - 21*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 56*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 105*sin(d*x + c)^4/(cos (d*x + c) + 1)^4 - 70*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 35*sin(d*x + c )^6/(cos(d*x + c) + 1)^6 + 1)/((a^3 + 6*a^3*sin(d*x + c)/(cos(d*x + c) + 1 ) + 14*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 14*a^3*sin(d*x + c)^3/(co s(d*x + c) + 1)^3 - 14*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 14*a^3*si n(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1 )^7 - a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d)
Time = 0.36 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.21 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {35}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} - \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 665 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 791 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 392 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 51}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/280*(35/(a^3*(tan(1/2*d*x + 1/2*c) - 1)) - (35*tan(1/2*d*x + 1/2*c)^6 - 280*tan(1/2*d*x + 1/2*c)^5 - 665*tan(1/2*d*x + 1/2*c)^4 - 1120*tan(1/2*d* x + 1/2*c)^3 - 791*tan(1/2*d*x + 1/2*c)^2 - 392*tan(1/2*d*x + 1/2*c) - 51) /(a^3*(tan(1/2*d*x + 1/2*c) + 1)^7))/d
Time = 31.87 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.08 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+56\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+70\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{35\,a^3\,d\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \] Input:
int(tan(c + d*x)/(cos(c + d*x)*(a + a*sin(c + d*x))^3),x)
Output:
(2*cos(c/2 + (d*x)/2)^2*(35*sin(c/2 + (d*x)/2)^6 - cos(c/2 + (d*x)/2)^6 + 70*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^5 - 6*cos(c/2 + (d*x)/2)^5*sin(c/ 2 + (d*x)/2) + 105*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 56*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^3 + 21*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x )/2)^2))/(35*a^3*d*(cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))*(cos(c/2 + (d *x)/2) + sin(c/2 + (d*x)/2))^7)
Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.41 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right )-\cos \left (d x +c \right )+6 \sin \left (d x +c \right )^{4}+18 \sin \left (d x +c \right )^{3}+15 \sin \left (d x +c \right )^{2}-3 \sin \left (d x +c \right )-1}{35 \cos \left (d x +c \right ) a^{3} d \left (\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1\right )} \] Input:
int(sec(d*x+c)*tan(d*x+c)/(a+a*sin(d*x+c))^3,x)
Output:
( - cos(c + d*x)*sin(c + d*x)**3 - 3*cos(c + d*x)*sin(c + d*x)**2 - 3*cos( c + d*x)*sin(c + d*x) - cos(c + d*x) + 6*sin(c + d*x)**4 + 18*sin(c + d*x) **3 + 15*sin(c + d*x)**2 - 3*sin(c + d*x) - 1)/(35*cos(c + d*x)*a**3*d*(si n(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1))