Integrand size = 27, antiderivative size = 85 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\sec (c+d x)}{a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {\sec ^5(c+d x)}{a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d} \] Output:
-sec(d*x+c)/a^2/d+4/3*sec(d*x+c)^3/a^2/d-sec(d*x+c)^5/a^2/d+2/7*sec(d*x+c) ^7/a^2/d-2/7*tan(d*x+c)^7/a^2/d
Time = 1.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.48 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\sec ^3(c+d x) (42-182 \cos (c+d x)+104 \cos (2 (c+d x))-39 \cos (3 (c+d x))-18 \cos (4 (c+d x))+13 \cos (5 (c+d x))+28 \sin (c+d x)-104 \sin (2 (c+d x))+66 \sin (3 (c+d x))-52 \sin (4 (c+d x))+6 \sin (5 (c+d x)))}{336 a^2 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]
Output:
-1/336*(Sec[c + d*x]^3*(42 - 182*Cos[c + d*x] + 104*Cos[2*(c + d*x)] - 39* Cos[3*(c + d*x)] - 18*Cos[4*(c + d*x)] + 13*Cos[5*(c + d*x)] + 28*Sin[c + d*x] - 104*Sin[2*(c + d*x)] + 66*Sin[3*(c + d*x)] - 52*Sin[4*(c + d*x)] + 6*Sin[5*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)
Time = 0.53 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5}{\cos (c+d x)^4 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \sec ^3(c+d x) (a-a \sin (c+d x))^2 \tan ^5(c+d x)dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^5 (a-a \sin (c+d x))^2}{\cos (c+d x)^8}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \sec (c+d x) \tan ^7(c+d x)-2 a^2 \sec ^2(c+d x) \tan ^6(c+d x)+a^2 \sec ^3(c+d x) \tan ^5(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 a^2 \tan ^7(c+d x)}{7 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}-\frac {a^2 \sec ^5(c+d x)}{d}+\frac {4 a^2 \sec ^3(c+d x)}{3 d}-\frac {a^2 \sec (c+d x)}{d}}{a^4}\) |
Input:
Int[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]
Output:
(-((a^2*Sec[c + d*x])/d) + (4*a^2*Sec[c + d*x]^3)/(3*d) - (a^2*Sec[c + d*x ]^5)/d + (2*a^2*Sec[c + d*x]^7)/(7*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 = 0.69 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(-\frac {2 \left (42 i {\mathrm e}^{8 i \left (d x +c \right )}+21 \,{\mathrm e}^{9 i \left (d x +c \right )}+56 i {\mathrm e}^{6 i \left (d x +c \right )}-28 \,{\mathrm e}^{7 i \left (d x +c \right )}+28 i {\mathrm e}^{4 i \left (d x +c \right )}-42 \,{\mathrm e}^{5 i \left (d x +c \right )}-24 i {\mathrm e}^{2 i \left (d x +c \right )}-76 \,{\mathrm e}^{3 i \left (d x +c \right )}-6 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{21 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} d \,a^{2}}\) | \(143\) |
| derivativedivides | \(\frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {64}{256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-256}+\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {5}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{2}}\) | \(145\) |
| default | \(\frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {64}{256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-256}+\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {5}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{2}}\) | \(145\) |
Input:
int(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2/21*(42*I*exp(8*I*(d*x+c))+21*exp(9*I*(d*x+c))+56*I*exp(6*I*(d*x+c))-28* exp(7*I*(d*x+c))+28*I*exp(4*I*(d*x+c))-42*exp(5*I*(d*x+c))-24*I*exp(2*I*(d *x+c))-76*exp(3*I*(d*x+c))-6*I-3*exp(I*(d*x+c)))/(exp(I*(d*x+c))-I)^3/(exp (I*(d*x+c))+I)^7/d/a^2
Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {9 \, \cos \left (d x + c\right )^{4} - 22 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 5}{21 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
-1/21*(9*cos(d*x + c)^4 - 22*cos(d*x + c)^2 - 2*(3*cos(d*x + c)^4 + 6*cos( d*x + c)^2 - 1)*sin(d*x + c) + 5)/(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)
\[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\sin {\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)**4/(a+a*sin(d*x+c))**2,x)
Output:
Integral(sin(c + d*x)*tan(c + d*x)**4/(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. 296 vs. \(2 (79) = 158\).
Time = 0.04 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.48 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {16 \, {\left (\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}}{21 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
-16/21*(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 - 14*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 + 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 = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.72 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {7 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1015 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1344 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 511 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 79}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{168 \, d} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/168*(7*(6*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) + 7)/(a^2*(ta n(1/2*d*x + 1/2*c) - 1)^3) - (42*tan(1/2*d*x + 1/2*c)^6 + 315*tan(1/2*d*x + 1/2*c)^5 + 1015*tan(1/2*d*x + 1/2*c)^4 + 1750*tan(1/2*d*x + 1/2*c)^3 + 1 344*tan(1/2*d*x + 1/2*c)^2 + 511*tan(1/2*d*x + 1/2*c) + 79)/(a^2*(tan(1/2* d*x + 1/2*c) + 1)^7))/d
Time = 34.59 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.88 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{21}+\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {128\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{21}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}}{a^2\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \] Input:
int((sin(c + d*x)*tan(c + d*x)^4)/(a + a*sin(c + d*x))^2,x)
Output:
-((16*cos(c/2 + (d*x)/2)^10)/21 + (64*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x) /2))/21 - (32*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^4)/3 - (128*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^3)/21 + (16*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^2)/7)/(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 = 0.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.69 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {14 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{3}-42 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )+5 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+28 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} \tan \left (d x +c \right )^{3}-84 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} \tan \left (d x +c \right )+10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-28 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \tan \left (d x +c \right )^{3}+84 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \tan \left (d x +c \right )-10 \cos \left (d x +c \right ) \sin \left (d x +c \right )-14 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{3}+42 \cos \left (d x +c \right ) \tan \left (d x +c \right )-5 \cos \left (d x +c \right )+68 \sin \left (d x +c \right )^{5}+94 \sin \left (d x +c \right )^{4}-34 \sin \left (d x +c \right )^{3}-92 \sin \left (d x +c \right )^{2}-10 \sin \left (d x +c \right )+16}{42 \cos \left (d x +c \right ) a^{2} d \left (\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )-1\right )} \] Input:
int(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^2,x)
Output:
(14*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x)**3 - 42*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x) + 5*cos(c + d*x)*sin(c + d*x)**4 + 28*cos(c + d*x)*s in(c + d*x)**3*tan(c + d*x)**3 - 84*cos(c + d*x)*sin(c + d*x)**3*tan(c + d *x) + 10*cos(c + d*x)*sin(c + d*x)**3 - 28*cos(c + d*x)*sin(c + d*x)*tan(c + d*x)**3 + 84*cos(c + d*x)*sin(c + d*x)*tan(c + d*x) - 10*cos(c + d*x)*s in(c + d*x) - 14*cos(c + d*x)*tan(c + d*x)**3 + 42*cos(c + d*x)*tan(c + d* x) - 5*cos(c + d*x) + 68*sin(c + d*x)**5 + 94*sin(c + d*x)**4 - 34*sin(c + d*x)**3 - 92*sin(c + d*x)**2 - 10*sin(c + d*x) + 16)/(42*cos(c + d*x)*a** 2*d*(sin(c + d*x)**4 + 2*sin(c + d*x)**3 - 2*sin(c + d*x) - 1))