Integrand size = 21, antiderivative size = 145 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {4 \sec ^5(c+d x)}{5 a^4 d}-\frac {16 \sec ^7(c+d x)}{7 a^4 d}+\frac {20 \sec ^9(c+d x)}{9 a^4 d}-\frac {8 \sec ^{11}(c+d x)}{11 a^4 d}+\frac {\tan ^5(c+d x)}{5 a^4 d}+\frac {9 \tan ^7(c+d x)}{7 a^4 d}+\frac {16 \tan ^9(c+d x)}{9 a^4 d}+\frac {8 \tan ^{11}(c+d x)}{11 a^4 d} \] Output:
4/5*sec(d*x+c)^5/a^4/d-16/7*sec(d*x+c)^7/a^4/d+20/9*sec(d*x+c)^9/a^4/d-8/1 1*sec(d*x+c)^11/a^4/d+1/5*tan(d*x+c)^5/a^4/d+9/7*tan(d*x+c)^7/a^4/d+16/9*t an(d*x+c)^9/a^4/d+8/11*tan(d*x+c)^11/a^4/d
Time = 1.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.14 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\sec ^3(c+d x) (168960-78903 \cos (c+d x)-183040 \cos (2 (c+d x))+8767 \cos (3 (c+d x))+62464 \cos (4 (c+d x))+19925 \cos (5 (c+d x))-15616 \cos (6 (c+d x))-797 \cos (7 (c+d x))+501600 \sin (c+d x)-70136 \sin (2 (c+d x))-200288 \sin (3 (c+d x))-25504 \sin (4 (c+d x))+48800 \sin (5 (c+d x))+6376 \sin (6 (c+d x))-1952 \sin (7 (c+d x)))}{3548160 a^4 d (1+\sin (c+d x))^4} \] Input:
Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]
Output:
(Sec[c + d*x]^3*(168960 - 78903*Cos[c + d*x] - 183040*Cos[2*(c + d*x)] + 8 767*Cos[3*(c + d*x)] + 62464*Cos[4*(c + d*x)] + 19925*Cos[5*(c + d*x)] - 1 5616*Cos[6*(c + d*x)] - 797*Cos[7*(c + d*x)] + 501600*Sin[c + d*x] - 70136 *Sin[2*(c + d*x)] - 200288*Sin[3*(c + d*x)] - 25504*Sin[4*(c + d*x)] + 488 00*Sin[5*(c + d*x)] + 6376*Sin[6*(c + d*x)] - 1952*Sin[7*(c + d*x)]))/(354 8160*a^4*d*(1 + Sin[c + d*x])^4)
Time = 0.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3190, 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.
| \(\displaystyle \int \frac {\tan ^4(c+d x)}{(a \sin (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^4}{(a \sin (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 3190 |
| \(\displaystyle \frac {\int \left (a^4 \tan ^4(c+d x) \sec ^8(c+d x)-4 a^4 \tan ^5(c+d x) \sec ^7(c+d x)+6 a^4 \tan ^6(c+d x) \sec ^6(c+d x)-4 a^4 \tan ^7(c+d x) \sec ^5(c+d x)+a^4 \tan ^8(c+d x) \sec ^4(c+d x)\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {8 a^4 \tan ^{11}(c+d x)}{11 d}+\frac {16 a^4 \tan ^9(c+d x)}{9 d}+\frac {9 a^4 \tan ^7(c+d x)}{7 d}+\frac {a^4 \tan ^5(c+d x)}{5 d}-\frac {8 a^4 \sec ^{11}(c+d x)}{11 d}+\frac {20 a^4 \sec ^9(c+d x)}{9 d}-\frac {16 a^4 \sec ^7(c+d x)}{7 d}+\frac {4 a^4 \sec ^5(c+d x)}{5 d}}{a^8}\) |
Input:
Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]
Output:
((4*a^4*Sec[c + d*x]^5)/(5*d) - (16*a^4*Sec[c + d*x]^7)/(7*d) + (20*a^4*Se c[c + d*x]^9)/(9*d) - (8*a^4*Sec[c + d*x]^11)/(11*d) + (a^4*Tan[c + d*x]^5 )/(5*d) + (9*a^4*Tan[c + d*x]^7)/(7*d) + (16*a^4*Tan[c + d*x]^9)/(9*d) + ( 8*a^4*Tan[c + d*x]^11)/(11*d))/a^8
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((g_.)*tan[(e_.) + (f_.)*(x _)])^(p_.), x_Symbol] :> Simp[a^(2*m) Int[ExpandIntegrand[(g*Tan[e + f*x] )^p/Sec[e + f*x]^m, (a*Sec[e + f*x] - b*Tan[e + f*x])^(-m), x], x], x] /; F reeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Result contains complex when optimal does not.
Time = 3.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(\frac {4 i \left (5544 i {\mathrm e}^{9 i \left (d x +c \right )}+3465 \,{\mathrm e}^{10 i \left (d x +c \right )}-5280 i {\mathrm e}^{7 i \left (d x +c \right )}-10857 \,{\mathrm e}^{8 i \left (d x +c \right )}+176 i {\mathrm e}^{5 i \left (d x +c \right )}+4818 \,{\mathrm e}^{6 i \left (d x +c \right )}-1952 i {\mathrm e}^{3 i \left (d x +c \right )}-2794 \,{\mathrm e}^{4 i \left (d x +c \right )}+488 i {\mathrm e}^{i \left (d x +c \right )}+1525 \,{\mathrm e}^{2 i \left (d x +c \right )}-61\right )}{3465 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{4}}\) | \(155\) |
| derivativedivides | \(\frac {-\frac {16}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {176}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {28}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {179}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {89}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {49}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{d \,a^{4}}\) | \(190\) |
| default | \(\frac {-\frac {16}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {176}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {28}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {179}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {89}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {49}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{d \,a^{4}}\) | \(190\) |
Input:
int(tan(d*x+c)^4/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
4/3465*I*(5544*I*exp(9*I*(d*x+c))+3465*exp(10*I*(d*x+c))-5280*I*exp(7*I*(d *x+c))-10857*exp(8*I*(d*x+c))+176*I*exp(5*I*(d*x+c))+4818*exp(6*I*(d*x+c)) -1952*I*exp(3*I*(d*x+c))-2794*exp(4*I*(d*x+c))+488*I*exp(I*(d*x+c))+1525*e xp(2*I*(d*x+c))-61)/(exp(I*(d*x+c))+I)^11/(exp(I*(d*x+c))-I)^3/d/a^4
Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {488 \, \cos \left (d x + c\right )^{6} - 1220 \, \cos \left (d x + c\right )^{4} + 1120 \, \cos \left (d x + c\right )^{2} + {\left (122 \, \cos \left (d x + c\right )^{6} - 915 \, \cos \left (d x + c\right )^{4} + 1400 \, \cos \left (d x + c\right )^{2} - 735\right )} \sin \left (d x + c\right ) - 420}{3465 \, {\left (a^{4} d \cos \left (d x + c\right )^{7} - 8 \, a^{4} d \cos \left (d x + c\right )^{5} + 8 \, a^{4} d \cos \left (d x + c\right )^{3} - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} - 2 \, a^{4} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(tan(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
-1/3465*(488*cos(d*x + c)^6 - 1220*cos(d*x + c)^4 + 1120*cos(d*x + c)^2 + (122*cos(d*x + c)^6 - 915*cos(d*x + c)^4 + 1400*cos(d*x + c)^2 - 735)*sin( d*x + c) - 420)/(a^4*d*cos(d*x + c)^7 - 8*a^4*d*cos(d*x + c)^5 + 8*a^4*d*c os(d*x + c)^3 - 4*(a^4*d*cos(d*x + c)^5 - 2*a^4*d*cos(d*x + c)^3)*sin(d*x + c))
\[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\int \frac {\tan ^{4}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(tan(d*x+c)**4/(a+a*sin(d*x+c))**4,x)
Output:
Integral(tan(c + d*x)**4/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a**4
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (129) = 258\).
Time = 0.04 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.37 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(tan(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
32/3465*(16*sin(d*x + c)/(cos(d*x + c) + 1) + 50*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 64*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 22*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 + 517*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 726*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1650*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 924*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 693*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 2)/((a^4 + 8*a^4*sin(d*x + c)/(cos(d*x + c) + 1) + 25*a^4*sin(d* x + c)^2/(cos(d*x + c) + 1)^2 + 32*a^4*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 11*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 88*a^4*sin(d*x + c)^5/(cos (d*x + c) + 1)^5 - 99*a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 99*a^4*sin (d*x + c)^8/(cos(d*x + c) + 1)^8 + 88*a^4*sin(d*x + c)^9/(cos(d*x + c) + 1 )^9 + 11*a^4*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 32*a^4*sin(d*x + c)^1 1/(cos(d*x + c) + 1)^11 - 25*a^4*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 8 *a^4*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - a^4*sin(d*x + c)^14/(cos(d*x + c) + 1)^14)*d)
Time = 0.33 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {1155 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 47355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 309540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 588588 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 891198 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 747450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 481140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 172700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35233 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3203}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{11}}}{110880 \, d} \] Input:
integrate(tan(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
-1/110880*(1155*(3*tan(1/2*d*x + 1/2*c) - 1)/(a^4*(tan(1/2*d*x + 1/2*c) - 1)^3) - (3465*tan(1/2*d*x + 1/2*c)^9 + 47355*tan(1/2*d*x + 1/2*c)^8 + 3095 40*tan(1/2*d*x + 1/2*c)^7 + 588588*tan(1/2*d*x + 1/2*c)^6 + 891198*tan(1/2 *d*x + 1/2*c)^5 + 747450*tan(1/2*d*x + 1/2*c)^4 + 481140*tan(1/2*d*x + 1/2 *c)^3 + 172700*tan(1/2*d*x + 1/2*c)^2 + 35233*tan(1/2*d*x + 1/2*c) + 3203) /(a^4*(tan(1/2*d*x + 1/2*c) + 1)^11))/d
Time = 36.74 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.92 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3465}+\frac {512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3465}+\frac {320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{693}+\frac {2048\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3465}-\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{315}+\frac {1504\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{315}+\frac {704\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{105}+\frac {320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{21}+\frac {128\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{15}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5}}{a^4\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^{11}} \] Input:
int(tan(c + d*x)^4/(a + a*sin(c + d*x))^4,x)
Output:
((64*cos(c/2 + (d*x)/2)^14)/3465 + (512*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d *x)/2))/3465 + (32*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9)/5 + (128*cos (c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8)/15 + (320*cos(c/2 + (d*x)/2)^7*sin (c/2 + (d*x)/2)^7)/21 + (704*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6)/10 5 + (1504*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5)/315 - (64*cos(c/2 + ( d*x)/2)^10*sin(c/2 + (d*x)/2)^4)/315 + (2048*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3)/3465 + (320*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2)/693 )/(a^4*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))^11)
Time = 0.18 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.53 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-128 \cos \left (d x +c \right ) \sin \left (d x +c \right )-32 \cos \left (d x +c \right )-122 \sin \left (d x +c \right )^{7}-488 \sin \left (d x +c \right )^{6}-549 \sin \left (d x +c \right )^{5}+244 \sin \left (d x +c \right )^{4}+64 \sin \left (d x +c \right )^{3}-144 \sin \left (d x +c \right )^{2}-128 \sin \left (d x +c \right )-32}{3465 \cos \left (d x +c \right ) a^{4} d \left (\sin \left (d x +c \right )^{6}+4 \sin \left (d x +c \right )^{5}+5 \sin \left (d x +c \right )^{4}-5 \sin \left (d x +c \right )^{2}-4 \sin \left (d x +c \right )-1\right )} \] Input:
int(tan(d*x+c)^4/(a+a*sin(d*x+c))^4,x)
Output:
(32*cos(c + d*x)*sin(c + d*x)**6 + 128*cos(c + d*x)*sin(c + d*x)**5 + 160* cos(c + d*x)*sin(c + d*x)**4 - 160*cos(c + d*x)*sin(c + d*x)**2 - 128*cos( c + d*x)*sin(c + d*x) - 32*cos(c + d*x) - 122*sin(c + d*x)**7 - 488*sin(c + d*x)**6 - 549*sin(c + d*x)**5 + 244*sin(c + d*x)**4 + 64*sin(c + d*x)**3 - 144*sin(c + d*x)**2 - 128*sin(c + d*x) - 32)/(3465*cos(c + d*x)*a**4*d* (sin(c + d*x)**6 + 4*sin(c + d*x)**5 + 5*sin(c + d*x)**4 - 5*sin(c + d*x)* *2 - 4*sin(c + d*x) - 1))