Integrand size = 21, antiderivative size = 127 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec ^3(c+d x)}{3 a^3 d}-\frac {6 \sec ^5(c+d x)}{5 a^3 d}+\frac {9 \sec ^7(c+d x)}{7 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {5 \tan ^7(c+d x)}{7 a^3 d}+\frac {4 \tan ^9(c+d x)}{9 a^3 d} \] Output:
1/3*sec(d*x+c)^3/a^3/d-6/5*sec(d*x+c)^5/a^3/d+9/7*sec(d*x+c)^7/a^3/d-4/9*s ec(d*x+c)^9/a^3/d+1/5*tan(d*x+c)^5/a^3/d+5/7*tan(d*x+c)^7/a^3/d+4/9*tan(d* x+c)^9/a^3/d
Time = 1.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.46 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {5376+1116 \cos (c+d x)-21312 \cos (2 (c+d x))+62 \cos (3 (c+d x))+8448 \cos (4 (c+d x))-186 \cos (5 (c+d x))-704 \cos (6 (c+d x))+39168 \sin (c+d x)+837 \sin (2 (c+d x))-28288 \sin (3 (c+d x))+372 \sin (4 (c+d x))+4224 \sin (5 (c+d x))-31 \sin (6 (c+d x))}{322560 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+a \sin (c+d x))^3} \] Input:
Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]
Output:
(5376 + 1116*Cos[c + d*x] - 21312*Cos[2*(c + d*x)] + 62*Cos[3*(c + d*x)] + 8448*Cos[4*(c + d*x)] - 186*Cos[5*(c + d*x)] - 704*Cos[6*(c + d*x)] + 391 68*Sin[c + d*x] + 837*Sin[2*(c + d*x)] - 28288*Sin[3*(c + d*x)] + 372*Sin[ 4*(c + d*x)] + 4224*Sin[5*(c + d*x)] - 31*Sin[6*(c + d*x)])/(322560*d*(Cos [(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 = 0.47 (sec) , antiderivative size = 131, 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.
\(\displaystyle \int \frac {\tan ^4(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)^4}{(a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3190 |
\(\displaystyle \frac {\int \left (-a^3 \sec ^3(c+d x) \tan ^7(c+d x)+3 a^3 \sec ^4(c+d x) \tan ^6(c+d x)-3 a^3 \sec ^5(c+d x) \tan ^5(c+d x)+a^3 \sec ^6(c+d x) \tan ^4(c+d x)\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {4 a^3 \tan ^9(c+d x)}{9 d}+\frac {5 a^3 \tan ^7(c+d x)}{7 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}-\frac {4 a^3 \sec ^9(c+d x)}{9 d}+\frac {9 a^3 \sec ^7(c+d x)}{7 d}-\frac {6 a^3 \sec ^5(c+d x)}{5 d}+\frac {a^3 \sec ^3(c+d x)}{3 d}}{a^6}\) |
Input:
Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]
Output:
((a^3*Sec[c + d*x]^3)/(3*d) - (6*a^3*Sec[c + d*x]^5)/(5*d) + (9*a^3*Sec[c + d*x]^7)/(7*d) - (4*a^3*Sec[c + d*x]^9)/(9*d) + (a^3*Tan[c + d*x]^5)/(5*d ) + (5*a^3*Tan[c + d*x]^7)/(7*d) + (4*a^3*Tan[c + d*x]^9)/(9*d))/a^6
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 = 1.72 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {\frac {88 \,{\mathrm e}^{i \left (d x +c \right )}}{105}+\frac {44 i}{315}+\frac {8 \,{\mathrm e}^{9 i \left (d x +c \right )}}{3}-\frac {176 i {\mathrm e}^{2 i \left (d x +c \right )}}{105}-\frac {32 \,{\mathrm e}^{7 i \left (d x +c \right )}}{5}+4 i {\mathrm e}^{8 i \left (d x +c \right )}+\frac {48 \,{\mathrm e}^{5 i \left (d x +c \right )}}{35}-\frac {16 i {\mathrm e}^{6 i \left (d x +c \right )}}{15}-\frac {928 \,{\mathrm e}^{3 i \left (d x +c \right )}}{315}+\frac {8 i {\mathrm e}^{4 i \left (d x +c \right )}}{35}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} a^{3} d}\) | \(143\) |
derivativedivides | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {32}{1024 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1024}-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {52}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {22}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {39}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{3} d}\) | \(175\) |
default | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {32}{1024 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1024}-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {52}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {22}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {39}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{3} d}\) | \(175\) |
Input:
int(tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
4/315*(66*exp(I*(d*x+c))+11*I+210*exp(9*I*(d*x+c))-132*I*exp(2*I*(d*x+c))- 504*exp(7*I*(d*x+c))+315*I*exp(8*I*(d*x+c))+108*exp(5*I*(d*x+c))-84*I*exp( 6*I*(d*x+c))-232*exp(3*I*(d*x+c))+18*I*exp(4*I*(d*x+c)))/(exp(I*(d*x+c))+I )^9/(exp(I*(d*x+c))-I)^3/a^3/d
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {22 \, \cos \left (d x + c\right )^{6} - 99 \, \cos \left (d x + c\right )^{4} + 120 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (33 \, \cos \left (d x + c\right )^{4} - 80 \, \cos \left (d x + c\right )^{2} + 35\right )} \sin \left (d x + c\right ) - 35}{315 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/315*(22*cos(d*x + c)^6 - 99*cos(d*x + c)^4 + 120*cos(d*x + c)^2 - 2*(33* cos(d*x + c)^4 - 80*cos(d*x + c)^2 + 35)*sin(d*x + c) - 35)/(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))
\[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\tan ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(tan(d*x+c)**4/(a+a*sin(d*x+c))**3,x)
Output:
Integral(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. 402 vs. \(2 (113) = 226\).
Time = 0.04 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.17 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {16 \, {\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {162 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {126 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {126 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 1\right )}}{315 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \] Input:
integrate(tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-16/315*(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 - 162*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 126*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 126*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 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/(c os(d*x + c) + 1)^8 - 2*a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 12*a^3*si n(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6*a^3*sin(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 = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.25 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {105 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1638 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8232 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2988 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 432 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 13}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \] Input:
integrate(tan(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/10080*(105*(3*tan(1/2*d*x + 1/2*c)^2 - 12*tan(1/2*d*x + 1/2*c) + 5)/(a^3 *(tan(1/2*d*x + 1/2*c) - 1)^3) - (315*tan(1/2*d*x + 1/2*c)^8 + 2520*tan(1/ 2*d*x + 1/2*c)^7 + 7140*tan(1/2*d*x + 1/2*c)^6 - 1638*tan(1/2*d*x + 1/2*c) ^4 - 8232*tan(1/2*d*x + 1/2*c)^3 - 2988*tan(1/2*d*x + 1/2*c)^2 - 432*tan(1 /2*d*x + 1/2*c) - 13)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d
Time = 36.69 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.83 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{315}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}+\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{315}-\frac {48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}-\frac {288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}}{a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^9} \] Input:
int(tan(c + d*x)^4/(a + a*sin(c + d*x))^3,x)
Output:
-((16*cos(c/2 + (d*x)/2)^12)/315 + (32*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d* x)/2))/105 - (32*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7)/5 - (32*cos(c/ 2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6)/5 - (288*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5)/35 - (48*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4)/35 + (3 2*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3)/315 + (64*cos(c/2 + (d*x)/2)^ 10*sin(c/2 + (d*x)/2)^2)/105)/(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 = 0.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.67 \[ \int \frac {\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )+8 \cos \left (d x +c \right )-22 \sin \left (d x +c \right )^{6}-66 \sin \left (d x +c \right )^{5}-33 \sin \left (d x +c \right )^{4}-28 \sin \left (d x +c \right )^{3}+12 \sin \left (d x +c \right )^{2}+24 \sin \left (d x +c \right )+8}{315 \cos \left (d x +c \right ) a^{3} d \left (\sin \left (d x +c \right )^{5}+3 \sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )^{2}-3 \sin \left (d x +c \right )-1\right )} \] Input:
int(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) - 22*sin(c + d*x)**6 - 66*sin(c + d*x )**5 - 33*sin(c + d*x)**4 - 28*sin(c + d*x)**3 + 12*sin(c + d*x)**2 + 24*s in(c + d*x) + 8)/(315*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))