Integrand size = 29, antiderivative size = 102 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec (c+d x)}{a^3 d}-\frac {2 \sec ^3(c+d x)}{a^3 d}+\frac {9 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d} \] Output:
sec(d*x+c)/a^3/d-2*sec(d*x+c)^3/a^3/d+9/5*sec(d*x+c)^5/a^3/d-4/7*sec(d*x+c )^7/a^3/d+1/5*tan(d*x+c)^5/a^3/d+4/7*tan(d*x+c)^7/a^3/d
Time = 2.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec (c+d x) (840-1946 \cos (c+d x)-224 \cos (2 (c+d x))+834 \cos (3 (c+d x))-104 \cos (4 (c+d x))+1344 \sin (c+d x)-1946 \sin (2 (c+d x))+64 \sin (3 (c+d x))+139 \sin (4 (c+d x)))}{2240 a^3 d (1+\sin (c+d x))^3} \] Input:
Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
Output:
(Sec[c + d*x]*(840 - 1946*Cos[c + d*x] - 224*Cos[2*(c + d*x)] + 834*Cos[3* (c + d*x)] - 104*Cos[4*(c + d*x)] + 1344*Sin[c + d*x] - 1946*Sin[2*(c + d* x)] + 64*Sin[3*(c + d*x)] + 139*Sin[4*(c + d*x)]))/(2240*a^3*d*(1 + Sin[c + d*x])^3)
Time = 0.62 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4}{\cos (c+d x)^2 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \sec ^4(c+d x) (a-a \sin (c+d x))^3 \tan ^4(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^4 (a-a \sin (c+d x))^3}{\cos (c+d x)^8}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (-a^3 \sec (c+d x) \tan ^7(c+d x)+3 a^3 \sec ^2(c+d x) \tan ^6(c+d x)-3 a^3 \sec ^3(c+d x) \tan ^5(c+d x)+a^3 \sec ^4(c+d x) \tan ^4(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {4 a^3 \tan ^7(c+d x)}{7 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}-\frac {4 a^3 \sec ^7(c+d x)}{7 d}+\frac {9 a^3 \sec ^5(c+d x)}{5 d}-\frac {2 a^3 \sec ^3(c+d x)}{d}+\frac {a^3 \sec (c+d x)}{d}}{a^6}\) |
Input:
Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
Output:
((a^3*Sec[c + d*x])/d - (2*a^3*Sec[c + d*x]^3)/d + (9*a^3*Sec[c + d*x]^5)/ (5*d) - (4*a^3*Sec[c + d*x]^7)/(7*d) + (a^3*Tan[c + d*x]^5)/(5*d) + (4*a^3 *Tan[c + d*x]^7)/(7*d))/a^6
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 = 1.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {-\frac {2 \,{\mathrm e}^{3 i \left (d x +c \right )}}{5}-\frac {22 i {\mathrm e}^{2 i \left (d x +c \right )}}{5}-6 i {\mathrm e}^{4 i \left (d x +c \right )}-10 \,{\mathrm e}^{5 i \left (d x +c \right )}+\frac {86 \,{\mathrm e}^{i \left (d x +c \right )}}{35}+\frac {26 i}{35}+6 i {\mathrm e}^{6 i \left (d x +c \right )}+2 \,{\mathrm e}^{7 i \left (d x +c \right )}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{3} d}\) | \(120\) |
| derivativedivides | \(\frac {-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {22}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {32}{256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+256}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{3} d}\) | \(130\) |
| default | \(\frac {-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {22}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {32}{256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+256}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{3} d}\) | \(130\) |
Input:
int(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
2/35*(-7*exp(3*I*(d*x+c))-77*I*exp(2*I*(d*x+c))-105*I*exp(4*I*(d*x+c))-175 *exp(5*I*(d*x+c))+43*exp(I*(d*x+c))+13*I+105*I*exp(6*I*(d*x+c))+35*exp(7*I *(d*x+c)))/(exp(I*(d*x+c))+I)^7/(exp(I*(d*x+c))-I)/a^3/d
Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + 5\right )} \sin \left (d x + c\right ) - 15}{35 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/35*(13*cos(d*x + c)^4 - 6*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + 5)*sin(d* x + c) - 15)/(3*a^3*d*cos(d*x + c)^3 - 4*a^3*d*cos(d*x + c) + (a^3*d*cos(d *x + c)^3 - 4*a^3*d*cos(d*x + c))*sin(d*x + c))
\[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )} \tan ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(sin(d*x+c)**2*tan(d*x+c)**2/(a+a*sin(d*x+c))**3,x)
Output:
Integral(sin(c + d*x)**2*tan(c + d*x)**2/(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. 230 vs. \(2 (94) = 188\).
Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.25 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {16 \, {\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 1\right )}}{35 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \] Input:
integrate(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
16/35*(6*sin(d*x + c)/(cos(d*x + c) + 1) + 14*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 14*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 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 + 1 4*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 14*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 14*a^3*sin(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 = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.18 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {35}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} - \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1015 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1673 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 616 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 93}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \] Input:
integrate(sin(d*x+c)^2*tan(d*x+c)^2/(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 + 1015*tan(1/2*d*x + 1/2*c)^4 + 2240*tan(1/2*d *x + 1/2*c)^3 + 1673*tan(1/2*d*x + 1/2*c)^2 + 616*tan(1/2*d*x + 1/2*c) + 9 3)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^7))/d
Time = 31.74 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{35}+\frac {96\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}}{a^3\,d\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \] Input:
int((sin(c + d*x)^2*tan(c + d*x)^2)/(a + a*sin(c + d*x))^3,x)
Output:
((16*cos(c/2 + (d*x)/2)^8)/35 + (96*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2 ))/35 + (32*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^3)/5 + (32*cos(c/2 + ( d*x)/2)^6*sin(c/2 + (d*x)/2)^2)/5)/(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 = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.37 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )+8 \cos \left (d x +c \right )-13 \sin \left (d x +c \right )^{4}-4 \sin \left (d x +c \right )^{3}+20 \sin \left (d x +c \right )^{2}+24 \sin \left (d x +c \right )+8}{35 \cos \left (d x +c \right ) a^{3} d \left (\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1\right )} \] Input:
int(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c))^3,x)
Output:
(8*cos(c + d*x)*sin(c + d*x)**3 + 24*cos(c + d*x)*sin(c + d*x)**2 + 24*cos (c + d*x)*sin(c + d*x) + 8*cos(c + d*x) - 13*sin(c + d*x)**4 - 4*sin(c + d *x)**3 + 20*sin(c + d*x)**2 + 24*sin(c + d*x) + 8)/(35*cos(c + d*x)*a**3*d *(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1))