Integrand size = 29, antiderivative size = 83 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}+\frac {\cos (c+d x)}{a d}+\frac {2 \sec (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d} \] Output:
x/a+cos(d*x+c)/a/d+2*sec(d*x+c)/a/d-1/3*sec(d*x+c)^3/a/d-tan(d*x+c)/a/d+1/ 3*tan(d*x+c)^3/a/d
Time = 1.50 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.78 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {18+2 (-11+6 c+6 d x) \cos (c+d x)+14 \cos (2 (c+d x))+11 \sin (c+d x)-11 \sin (2 (c+d x))+6 c \sin (2 (c+d x))+6 d x \sin (2 (c+d x))+3 \sin (3 (c+d x))}{12 a d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (1+\sin (c+d x))} \] Input:
Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
(18 + 2*(-11 + 6*c + 6*d*x)*Cos[c + d*x] + 14*Cos[2*(c + d*x)] + 11*Sin[c + d*x] - 11*Sin[2*(c + d*x)] + 6*c*Sin[2*(c + d*x)] + 6*d*x*Sin[2*(c + d*x )] + 3*Sin[3*(c + d*x)])/(12*a*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Co s[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))
Time = 0.49 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3318, 3042, 3070, 244, 2009, 3954, 3042, 3954, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4}{\cos (c+d x)^2 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \tan ^4(c+d x)dx}{a}-\frac {\int \sin (c+d x) \tan ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \tan (c+d x)^4dx}{a}-\frac {\int \sin (c+d x) \tan (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 3070 |
| \(\displaystyle \frac {\int \tan (c+d x)^4dx}{a}+\frac {\int \left (1-\cos ^2(c+d x)\right )^2 \sec ^4(c+d x)d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \tan (c+d x)^4dx}{a}+\frac {\int \left (\sec ^4(c+d x)-2 \sec ^2(c+d x)+1\right )d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \tan (c+d x)^4dx}{a}+\frac {\cos (c+d x)-\frac {1}{3} \sec ^3(c+d x)+2 \sec (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\frac {\tan ^3(c+d x)}{3 d}-\int \tan ^2(c+d x)dx}{a}+\frac {\cos (c+d x)-\frac {1}{3} \sec ^3(c+d x)+2 \sec (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\tan ^3(c+d x)}{3 d}-\int \tan (c+d x)^2dx}{a}+\frac {\cos (c+d x)-\frac {1}{3} \sec ^3(c+d x)+2 \sec (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\int 1dx+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}}{a}+\frac {\cos (c+d x)-\frac {1}{3} \sec ^3(c+d x)+2 \sec (c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}+x}{a}+\frac {\cos (c+d x)-\frac {1}{3} \sec ^3(c+d x)+2 \sec (c+d x)}{a d}\) |
Input:
Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
(Cos[c + d*x] + 2*Sec[c + d*x] - Sec[c + d*x]^3/3)/(a*d) + (x - Tan[c + d* x]/d + Tan[c + d*x]^3/(3*d))/a
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Time = 0.56 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {32}{16+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(99\) |
| default | \(\frac {-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {32}{16+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(99\) |
| risch | \(\frac {x}{a}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {\frac {4 \,{\mathrm e}^{i \left (d x +c \right )}}{3}+4 i {\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{3 i \left (d x +c \right )}+\frac {8 i}{3}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} d a}\) | \(112\) |
Input:
int(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
32/d/a*(-1/64/(tan(1/2*d*x+1/2*c)-1)+1/16/(1+tan(1/2*d*x+1/2*c)^2)+1/16*ar ctan(tan(1/2*d*x+1/2*c))-1/48/(tan(1/2*d*x+1/2*c)+1)^3+1/32/(tan(1/2*d*x+1 /2*c)+1)^2+5/64/(tan(1/2*d*x+1/2*c)+1))
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, d x \cos \left (d x + c\right ) + 7 \, \cos \left (d x + c\right )^{2} + {\left (3 \, d x \cos \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 1}{3 \, {\left (a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )\right )}} \] Input:
integrate(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/3*(3*d*x*cos(d*x + c) + 7*cos(d*x + c)^2 + (3*d*x*cos(d*x + c) + 3*cos(d *x + c)^2 + 2)*sin(d*x + c) + 1)/(a*d*cos(d*x + c)*sin(d*x + c) + a*d*cos( d*x + c))
\[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )} \tan ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(sin(d*x+c)**2*tan(d*x+c)**2/(a+a*sin(d*x+c)),x)
Output:
Integral(sin(c + d*x)**2*tan(c + d*x)**2/(sin(c + d*x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (79) = 158\).
Time = 0.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.84 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \, {\left (\frac {\frac {13 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 8}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{3 \, d} \] Input:
integrate(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
2/3*((13*sin(d*x + c)/(cos(d*x + c) + 1) + 2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 8)/(a + 2*a*sin(d*x + c)/(cos(d*x + c) + 1) + a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - a*sin(d *x + c)^4/(cos(d*x + c) + 1)^4 - 2*a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.51 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {6 \, {\left (d x + c\right )}}{a} - \frac {3 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} a} + \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \] Input:
integrate(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/6*(6*(d*x + c)/a - 3*(tan(1/2*d*x + 1/2*c)^2 - 4*tan(1/2*d*x + 1/2*c) + 5)/((tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c ) - 1)*a) + (15*tan(1/2*d*x + 1/2*c)^2 + 36*tan(1/2*d*x + 1/2*c) + 17)/(a* (tan(1/2*d*x + 1/2*c) + 1)^3))/d
Time = 33.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.55 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {26\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {16}{3}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \] Input:
int((sin(c + d*x)^2*tan(c + d*x)^2)/(a + a*sin(c + d*x)),x)
Output:
x/a - ((26*tan(c/2 + (d*x)/2))/3 + (4*tan(c/2 + (d*x)/2)^2)/3 - (4*tan(c/2 + (d*x)/2)^3)/3 - 4*tan(c/2 + (d*x)/2)^4 - 2*tan(c/2 + (d*x)/2)^5 + 16/3) /(a*d*(tan(c/2 + (d*x)/2) + 1)^3*(tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^ 2 + tan(c/2 + (d*x)/2)^3 - 1))
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.54 \[ \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d x +5 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 \cos \left (d x +c \right ) c +3 \cos \left (d x +c \right ) d x +5 \cos \left (d x +c \right )-3 \sin \left (d x +c \right )^{3}-7 \sin \left (d x +c \right )^{2}+5 \sin \left (d x +c \right )+8}{3 \cos \left (d x +c \right ) a d \left (\sin \left (d x +c \right )+1\right )} \] Input:
int(sin(d*x+c)^2*tan(d*x+c)^2/(a+a*sin(d*x+c)),x)
Output:
(3*cos(c + d*x)*sin(c + d*x)*c + 3*cos(c + d*x)*sin(c + d*x)*d*x + 5*cos(c + d*x)*sin(c + d*x) + 3*cos(c + d*x)*c + 3*cos(c + d*x)*d*x + 5*cos(c + d *x) - 3*sin(c + d*x)**3 - 7*sin(c + d*x)**2 + 5*sin(c + d*x) + 8)/(3*cos(c + d*x)*a*d*(sin(c + d*x) + 1))