Integrand size = 27, antiderivative size = 105 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\sec ^3(c+d x)}{a^3 d}+\frac {2 \sec ^5(c+d x)}{a^3 d}-\frac {11 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {3 \tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d} \] Output:
-sec(d*x+c)^3/a^3/d+2*sec(d*x+c)^5/a^3/d-11/7*sec(d*x+c)^7/a^3/d+4/9*sec(d *x+c)^9/a^3/d-3/7*tan(d*x+c)^7/a^3/d-4/9*tan(d*x+c)^9/a^3/d
Time = 1.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.76 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-1344+8676 \cos (c+d x)-11232 \cos (2 (c+d x))+482 \cos (3 (c+d x))+4416 \cos (4 (c+d x))-1446 \cos (5 (c+d x))-32 \cos (6 (c+d x))-1152 \sin (c+d x)+6507 \sin (2 (c+d x))-8128 \sin (3 (c+d x))+2892 \sin (4 (c+d x))+192 \sin (5 (c+d x))-241 \sin (6 (c+d x))}{64512 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+a \sin (c+d x))^3} \] Input:
Integrate[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]
Output:
(-1344 + 8676*Cos[c + d*x] - 11232*Cos[2*(c + d*x)] + 482*Cos[3*(c + d*x)] + 4416*Cos[4*(c + d*x)] - 1446*Cos[5*(c + d*x)] - 32*Cos[6*(c + d*x)] - 1 152*Sin[c + d*x] + 6507*Sin[2*(c + d*x)] - 8128*Sin[3*(c + d*x)] + 2892*Si n[4*(c + d*x)] + 192*Sin[5*(c + d*x)] - 241*Sin[6*(c + d*x)])/(64512*d*(Co s[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) ^3*(a + a*Sin[c + d*x])^3)
Time = 0.60 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5}{\cos (c+d x)^4 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \sec ^5(c+d x) (a-a \sin (c+d x))^3 \tan ^5(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^5 (a-a \sin (c+d x))^3}{\cos (c+d x)^{10}}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (-a^3 \sec ^2(c+d x) \tan ^8(c+d x)+3 a^3 \sec ^3(c+d x) \tan ^7(c+d x)-3 a^3 \sec ^4(c+d x) \tan ^6(c+d x)+a^3 \sec ^5(c+d x) \tan ^5(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {4 a^3 \tan ^9(c+d x)}{9 d}-\frac {3 a^3 \tan ^7(c+d x)}{7 d}+\frac {4 a^3 \sec ^9(c+d x)}{9 d}-\frac {11 a^3 \sec ^7(c+d x)}{7 d}+\frac {2 a^3 \sec ^5(c+d x)}{d}-\frac {a^3 \sec ^3(c+d x)}{d}}{a^6}\) |
Input:
Int[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]
Output:
(-((a^3*Sec[c + d*x]^3)/d) + (2*a^3*Sec[c + d*x]^5)/d - (11*a^3*Sec[c + d* x]^7)/(7*d) + (4*a^3*Sec[c + d*x]^9)/(9*d) - (3*a^3*Tan[c + d*x]^7)/(7*d) - (4*a^3*Tan[c + d*x]^9)/(9*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 = 1.25 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.36
| method | result | size |
| risch | \(-\frac {2 i \left (-128 i {\mathrm e}^{3 i \left (d x +c \right )}+75 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i {\mathrm e}^{i \left (d x +c \right )}-162 \,{\mathrm e}^{4 i \left (d x +c \right )}-1-36 i {\mathrm e}^{5 i \left (d x +c \right )}-42 \,{\mathrm e}^{6 i \left (d x +c \right )}-189 \,{\mathrm e}^{8 i \left (d x +c \right )}+126 i {\mathrm e}^{9 i \left (d x +c \right )}+63 \,{\mathrm e}^{10 i \left (d x +c \right )}\right )}{63 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} d \,a^{3}}\) | \(143\) |
| derivativedivides | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {48}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {16}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{3} d}\) | \(190\) |
| default | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {48}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {16}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{3} d}\) | \(190\) |
Input:
int(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/63*I*(-128*I*exp(3*I*(d*x+c))+75*exp(2*I*(d*x+c))+6*I*exp(I*(d*x+c))-16 2*exp(4*I*(d*x+c))-1-36*I*exp(5*I*(d*x+c))-42*exp(6*I*(d*x+c))-189*exp(8*I *(d*x+c))+126*I*exp(9*I*(d*x+c))+63*exp(10*I*(d*x+c)))/(exp(I*(d*x+c))-I)^ 3/(exp(I*(d*x+c))+I)^9/d/a^3
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.22 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos \left (d x + c\right )^{6} - 36 \, \cos \left (d x + c\right )^{4} + 57 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 14}{63 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/63*(cos(d*x + c)^6 - 36*cos(d*x + c)^4 + 57*cos(d*x + c)^2 - (3*cos(d*x + c)^4 - 34*cos(d*x + c)^2 + 7)*sin(d*x + c) - 14)/(3*a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3 + (a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3 )*sin(d*x + c))
\[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sin {\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)**4/(a+a*sin(d*x+c))**3,x)
Output:
Integral(sin(c + d*x)*tan(c + d*x)**4/(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. 382 vs. \(2 (97) = 194\).
Time = 0.04 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.64 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {16 \, {\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}}{63 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-16/63*(6*sin(d*x + c)/(cos(d*x + c) + 1) + 12*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 + 2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 27*sin(d*x + c)^4/(cos( d*x + c) + 1)^4 - 36*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 42*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) + 12*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2*a^3*sin(d*x + c)^3/(cos( d*x + c) + 1)^3 - 27*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*a^3*sin( d*x + c)^5/(cos(d*x + c) + 1)^5 + 36*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1) ^7 + 27*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 2*a^3*sin(d*x + c)^9/(co s(d*x + c) + 1)^9 - 12*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6*a^3*s in(d*x + c)^11/(cos(d*x + c) + 1)^11 - a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12)*d)
Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.64 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {21 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {189 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1764 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7224 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 16380 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 19026 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16380 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8352 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2340 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 281}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \] Input:
integrate(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/2016*(21*(9*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c) + 11)/(a^3* (tan(1/2*d*x + 1/2*c) - 1)^3) - (189*tan(1/2*d*x + 1/2*c)^8 + 1764*tan(1/2 *d*x + 1/2*c)^7 + 7224*tan(1/2*d*x + 1/2*c)^6 + 16380*tan(1/2*d*x + 1/2*c) ^5 + 19026*tan(1/2*d*x + 1/2*c)^4 + 16380*tan(1/2*d*x + 1/2*c)^3 + 8352*ta n(1/2*d*x + 1/2*c)^2 + 2340*tan(1/2*d*x + 1/2*c) + 281)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d
Time = 35.02 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.98 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{63}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}+\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{21}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{63}-\frac {48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{7}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}}{a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^9} \] Input:
int((sin(c + d*x)*tan(c + d*x)^4)/(a + a*sin(c + d*x))^3,x)
Output:
-((16*cos(c/2 + (d*x)/2)^12)/63 + (32*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x )/2))/21 - (32*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6)/3 - (64*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5)/7 - (48*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4)/7 + (32*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3)/63 + (64*co s(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2)/21)/(a^3*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))^9)
Time = 0.18 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.02 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )+8 \cos \left (d x +c \right )-\sin \left (d x +c \right )^{6}-3 \sin \left (d x +c \right )^{5}-33 \sin \left (d x +c \right )^{4}-28 \sin \left (d x +c \right )^{3}+12 \sin \left (d x +c \right )^{2}+24 \sin \left (d x +c \right )+8}{63 \cos \left (d x +c \right ) a^{3} d \left (\sin \left (d x +c \right )^{5}+3 \sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )^{2}-3 \sin \left (d x +c \right )-1\right )} \] Input:
int(sin(d*x+c)*tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x)
Output:
( - 8*cos(c + d*x)*sin(c + d*x)**5 - 24*cos(c + d*x)*sin(c + d*x)**4 - 16* cos(c + d*x)*sin(c + d*x)**3 + 16*cos(c + d*x)*sin(c + d*x)**2 + 24*cos(c + d*x)*sin(c + d*x) + 8*cos(c + d*x) - sin(c + d*x)**6 - 3*sin(c + d*x)**5 - 33*sin(c + d*x)**4 - 28*sin(c + d*x)**3 + 12*sin(c + d*x)**2 + 24*sin(c + d*x) + 8)/(63*cos(c + d*x)*a**3*d*(sin(c + d*x)**5 + 3*sin(c + d*x)**4 + 2*sin(c + d*x)**3 - 2*sin(c + d*x)**2 - 3*sin(c + d*x) - 1))