Integrand size = 26, antiderivative size = 76 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f} \] Output:
1/5*sec(f*x+e)^3/a^2/f/(c^3-c^3*sin(f*x+e))+4/5*tan(f*x+e)/a^2/c^3/f+4/15* tan(f*x+e)^3/a^2/c^3/f
Time = 1.50 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {1050 \cos (e+f x)+256 \cos (2 (e+f x))+350 \cos (3 (e+f x))+128 \cos (4 (e+f x))+768 \sin (e+f x)-350 \sin (2 (e+f x))+256 \sin (3 (e+f x))-175 \sin (4 (e+f x))}{1920 a^2 c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:
Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3),x]
Output:
(1050*Cos[e + f*x] + 256*Cos[2*(e + f*x)] + 350*Cos[3*(e + f*x)] + 128*Cos [4*(e + f*x)] + 768*Sin[e + f*x] - 350*Sin[2*(e + f*x)] + 256*Sin[3*(e + f *x)] - 175*Sin[4*(e + f*x)])/(1920*a^2*c^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)
Time = 0.41 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3042, 3215, 3042, 3151, 3042, 4254, 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.
\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^3}dx\) |
\(\Big \downarrow \) 3215 |
\(\displaystyle \frac {\int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)}dx}{a^2 c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))}dx}{a^2 c^2}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {\frac {4 \int \sec ^4(e+f x)dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}}{a^2 c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {4 \int \csc \left (e+f x+\frac {\pi }{2}\right )^4dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}}{a^2 c^2}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \int \left (\tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{5 c f}}{a^2 c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \left (-\frac {1}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{5 c f}}{a^2 c^2}\) |
Input:
Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3),x]
Output:
(Sec[e + f*x]^3/(5*f*(c - c*Sin[e + f*x])) - (4*(-Tan[e + f*x] - Tan[e + f *x]^3/3))/(5*c*f))/(a^2*c^2)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {\frac {32 \,{\mathrm e}^{3 i \left (f x +e \right )}}{5}-\frac {32 i {\mathrm e}^{2 i \left (f x +e \right )}}{15}+\frac {32 \,{\mathrm e}^{i \left (f x +e \right )}}{15}-\frac {16 i}{15}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{3} a^{2}}\) | \(77\) |
parallelrisch | \(\frac {-\frac {2}{5}-\frac {26 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{15}-\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3}+2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}+\frac {14 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{5}}{f \,c^{3} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(129\) |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {5}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {5}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{3} f}\) | \(133\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {5}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {5}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{3} f}\) | \(133\) |
norman | \(\frac {-\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 a f c}-\frac {2}{5 a c f}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a f c}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f c}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5 a c f}+\frac {14 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{5 a c f}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f c}-\frac {26 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{15 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(198\) |
Input:
int(1/(a+sin(f*x+e)*a)^2/(c-c*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
16/15*(6*exp(3*I*(f*x+e))-2*I*exp(2*I*(f*x+e))+2*exp(I*(f*x+e))-I)/(exp(I* (f*x+e))-I)^5/(exp(I*(f*x+e))+I)^3/f/c^3/a^2
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {8 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sin \left (f x + e\right ) - 1}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a^{2} c^{3} f \cos \left (f x + e\right )^{3}\right )}} \] Input:
integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-1/15*(8*cos(f*x + e)^4 - 4*cos(f*x + e)^2 + 4*(2*cos(f*x + e)^2 + 1)*sin( f*x + e) - 1)/(a^2*c^3*f*cos(f*x + e)^3*sin(f*x + e) - a^2*c^3*f*cos(f*x + e)^3)
Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (66) = 132\).
Time = 5.04 (sec) , antiderivative size = 1418, normalized size of antiderivative = 18.66 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**3,x)
Output:
Piecewise((-30*tan(e/2 + f*x/2)**7/(15*a**2*c**3*f*tan(e/2 + f*x/2)**8 - 3 0*a**2*c**3*f*tan(e/2 + f*x/2)**7 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**6 + 9 0*a**2*c**3*f*tan(e/2 + f*x/2)**5 - 90*a**2*c**3*f*tan(e/2 + f*x/2)**3 + 3 0*a**2*c**3*f*tan(e/2 + f*x/2)**2 + 30*a**2*c**3*f*tan(e/2 + f*x/2) - 15*a **2*c**3*f) + 30*tan(e/2 + f*x/2)**6/(15*a**2*c**3*f*tan(e/2 + f*x/2)**8 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**7 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**6 + 90*a**2*c**3*f*tan(e/2 + f*x/2)**5 - 90*a**2*c**3*f*tan(e/2 + f*x/2)**3 + 30*a**2*c**3*f*tan(e/2 + f*x/2)**2 + 30*a**2*c**3*f*tan(e/2 + f*x/2) - 15 *a**2*c**3*f) + 10*tan(e/2 + f*x/2)**5/(15*a**2*c**3*f*tan(e/2 + f*x/2)**8 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**7 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**6 + 90*a**2*c**3*f*tan(e/2 + f*x/2)**5 - 90*a**2*c**3*f*tan(e/2 + f*x/2)**3 + 30*a**2*c**3*f*tan(e/2 + f*x/2)**2 + 30*a**2*c**3*f*tan(e/2 + f*x/2) - 15*a**2*c**3*f) - 50*tan(e/2 + f*x/2)**4/(15*a**2*c**3*f*tan(e/2 + f*x/2)* *8 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**7 - 30*a**2*c**3*f*tan(e/2 + f*x/2)* *6 + 90*a**2*c**3*f*tan(e/2 + f*x/2)**5 - 90*a**2*c**3*f*tan(e/2 + f*x/2)* *3 + 30*a**2*c**3*f*tan(e/2 + f*x/2)**2 + 30*a**2*c**3*f*tan(e/2 + f*x/2) - 15*a**2*c**3*f) - 26*tan(e/2 + f*x/2)**3/(15*a**2*c**3*f*tan(e/2 + f*x/2 )**8 - 30*a**2*c**3*f*tan(e/2 + f*x/2)**7 - 30*a**2*c**3*f*tan(e/2 + f*x/2 )**6 + 90*a**2*c**3*f*tan(e/2 + f*x/2)**5 - 90*a**2*c**3*f*tan(e/2 + f*x/2 )**3 + 30*a**2*c**3*f*tan(e/2 + f*x/2)**2 + 30*a**2*c**3*f*tan(e/2 + f*...
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (71) = 142\).
Time = 0.04 (sec) , antiderivative size = 335, normalized size of antiderivative = 4.41 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {2 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {13 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {25 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 3\right )}}{15 \, {\left (a^{2} c^{3} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a^{2} c^{3} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} f} \] Input:
integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^3,x, algorithm="maxima")
Output:
2/15*(9*sin(f*x + e)/(cos(f*x + e) + 1) - 21*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 13*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 25*sin(f*x + e)^4/(cos(f *x + e) + 1)^4 - 5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 15*sin(f*x + e)^6 /(cos(f*x + e) + 1)^6 + 15*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3)/((a^2* c^3 - 2*a^2*c^3*sin(f*x + e)/(cos(f*x + e) + 1) - 2*a^2*c^3*sin(f*x + e)^2 /(cos(f*x + e) + 1)^2 + 6*a^2*c^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6* a^2*c^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 2*a^2*c^3*sin(f*x + e)^6/(co s(f*x + e) + 1)^6 + 2*a^2*c^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - a^2*c^ 3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)*f)
Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {\frac {5 \, {\left (15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13\right )}}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {165 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 480 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 650 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 400 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 113}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{120 \, f} \] Input:
integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^3,x, algorithm="giac")
Output:
-1/120*(5*(15*tan(1/2*f*x + 1/2*e)^2 + 24*tan(1/2*f*x + 1/2*e) + 13)/(a^2* c^3*(tan(1/2*f*x + 1/2*e) + 1)^3) + (165*tan(1/2*f*x + 1/2*e)^4 - 480*tan( 1/2*f*x + 1/2*e)^3 + 650*tan(1/2*f*x + 1/2*e)^2 - 400*tan(1/2*f*x + 1/2*e) + 113)/(a^2*c^3*(tan(1/2*f*x + 1/2*e) - 1)^5))/f
Time = 18.38 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+25\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+13\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+9\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+3\right )}{15\,a^2\,c^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^5\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \] Input:
int(1/((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^3),x)
Output:
-(2*(9*tan(e/2 + (f*x)/2) - 21*tan(e/2 + (f*x)/2)^2 + 13*tan(e/2 + (f*x)/2 )^3 + 25*tan(e/2 + (f*x)/2)^4 - 5*tan(e/2 + (f*x)/2)^5 - 15*tan(e/2 + (f*x )/2)^6 + 15*tan(e/2 + (f*x)/2)^7 + 3))/(15*a^2*c^3*f*(tan(e/2 + (f*x)/2) - 1)^5*(tan(e/2 + (f*x)/2) + 1)^3)
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {-12 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+12 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+12 \cos \left (f x +e \right ) \sin \left (f x +e \right )-12 \cos \left (f x +e \right )+8 \sin \left (f x +e \right )^{4}-8 \sin \left (f x +e \right )^{3}-12 \sin \left (f x +e \right )^{2}+12 \sin \left (f x +e \right )+3}{15 \cos \left (f x +e \right ) a^{2} c^{3} f \left (\sin \left (f x +e \right )^{3}-\sin \left (f x +e \right )^{2}-\sin \left (f x +e \right )+1\right )} \] Input:
int(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^3,x)
Output:
( - 12*cos(e + f*x)*sin(e + f*x)**3 + 12*cos(e + f*x)*sin(e + f*x)**2 + 12 *cos(e + f*x)*sin(e + f*x) - 12*cos(e + f*x) + 8*sin(e + f*x)**4 - 8*sin(e + f*x)**3 - 12*sin(e + f*x)**2 + 12*sin(e + f*x) + 3)/(15*cos(e + f*x)*a* *2*c**3*f*(sin(e + f*x)**3 - sin(e + f*x)**2 - sin(e + f*x) + 1))