Integrand size = 26, antiderivative size = 144 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f} \]
1/9*sec(f*x+e)^3/a^2/c^2/f/(c-c*sin(f*x+e))^3+2/21*sec(f*x+e)^3/a^2/c^3/f/ (c-c*sin(f*x+e))^2+2/21*sec(f*x+e)^3/a^2/f/(c^5-c^5*sin(f*x+e))+8/21*tan(f *x+e)/a^2/c^5/f+8/63*tan(f*x+e)^3/a^2/c^5/f
Time = 2.06 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (2397276 \cos (e+f x)+221184 \cos (2 (e+f x))+133182 \cos (3 (e+f x))+98304 \cos (4 (e+f x))-399546 \cos (5 (e+f x))-8192 \cos (6 (e+f x))+294912 \sin (e+f x)-1797957 \sin (2 (e+f x))+16384 \sin (3 (e+f x))-799092 \sin (4 (e+f x))-49152 \sin (5 (e+f x))+66591 \sin (6 (e+f x)))}{1032192 a^2 c^5 f (-1+\sin (e+f x))^5 (1+\sin (e+f x))^2} \]
-1/1032192*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[ (e + f*x)/2])*(2397276*Cos[e + f*x] + 221184*Cos[2*(e + f*x)] + 133182*Cos [3*(e + f*x)] + 98304*Cos[4*(e + f*x)] - 399546*Cos[5*(e + f*x)] - 8192*Co s[6*(e + f*x)] + 294912*Sin[e + f*x] - 1797957*Sin[2*(e + f*x)] + 16384*Si n[3*(e + f*x)] - 799092*Sin[4*(e + f*x)] - 49152*Sin[5*(e + f*x)] + 66591* Sin[6*(e + f*x)]))/(a^2*c^5*f*(-1 + Sin[e + f*x])^5*(1 + Sin[e + f*x])^2)
Time = 0.68 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 3215, 3042, 3151, 3042, 3151, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^3}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))^3}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {2 \int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2}dx}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))^2}dx}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {2 \left (\frac {5 \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)}dx}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (\frac {5 \int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))}dx}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {2 \left (\frac {5 \left (\frac {4 \int \sec ^4(e+f x)dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (\frac {5 \left (\frac {4 \int \csc \left (e+f x+\frac {\pi }{2}\right )^4dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {2 \left (\frac {5 \left (\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \int \left (\tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{5 c f}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}+\frac {2 \left (\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}+\frac {5 \left (\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \left (-\frac {1}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{5 c f}\right )}{7 c}\right )}{3 c}}{a^2 c^2}\) |
(Sec[e + f*x]^3/(9*f*(c - c*Sin[e + f*x])^3) + (2*(Sec[e + f*x]^3/(7*f*(c - c*Sin[e + f*x])^2) + (5*(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)))/(7*c)))/(3*c))/(a^2*c^2)
3.3.78.3.1 Defintions of rubi rules used
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 = 2.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {32 \left (i+36 \,{\mathrm e}^{5 i \left (f x +e \right )}-12 i {\mathrm e}^{2 i \left (f x +e \right )}-27 i {\mathrm e}^{4 i \left (f x +e \right )}-6 \,{\mathrm e}^{i \left (f x +e \right )}+2 \,{\mathrm e}^{3 i \left (f x +e \right )}\right )}{63 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{5} a^{2}}\) | \(100\) |
| parallelrisch | \(\frac {-\frac {38}{63}+6 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {68 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {26 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {100 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-\frac {28 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+12 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {470 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21}-\frac {26 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21}}{f \,c^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(181\) |
| derivativedivides | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {59}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {9}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{5} f}\) | \(193\) |
| default | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {59}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {9}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{5} f}\) | \(193\) |
| norman | \(\frac {-\frac {28 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {38}{63 a c f}+\frac {6 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {12 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {26 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {470 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 a c f}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 a c f}-\frac {68 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {26 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 a c f}+\frac {100 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(286\) |
-32/63*(I+36*exp(5*I*(f*x+e))-12*I*exp(2*I*(f*x+e))-27*I*exp(4*I*(f*x+e))- 6*exp(I*(f*x+e))+2*exp(3*I*(f*x+e)))/(exp(I*(f*x+e))-I)^9/(exp(I*(f*x+e))+ I)^3/f/c^5/a^2
Time = 0.34 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {16 \, \cos \left (f x + e\right )^{6} - 72 \, \cos \left (f x + e\right )^{4} + 30 \, \cos \left (f x + e\right )^{2} + 2 \, {\left (24 \, \cos \left (f x + e\right )^{4} - 20 \, \cos \left (f x + e\right )^{2} - 7\right )} \sin \left (f x + e\right ) + 7}{63 \, {\left (3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} - {\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )}} \]
1/63*(16*cos(f*x + e)^6 - 72*cos(f*x + e)^4 + 30*cos(f*x + e)^2 + 2*(24*co s(f*x + e)^4 - 20*cos(f*x + e)^2 - 7)*sin(f*x + e) + 7)/(3*a^2*c^5*f*cos(f *x + e)^5 - 4*a^2*c^5*f*cos(f*x + e)^3 - (a^2*c^5*f*cos(f*x + e)^5 - 4*a^2 *c^5*f*cos(f*x + e)^3)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 3186 vs. \(2 (131) = 262\).
Time = 20.31 (sec) , antiderivative size = 3186, normalized size of antiderivative = 22.12 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]
Piecewise((-126*tan(e/2 + f*x/2)**11/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)* *10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x /2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/ 2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan( e/2 + f*x/2) - 63*a**2*c**5*f) + 378*tan(e/2 + f*x/2)**10/(63*a**2*c**5*f* tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c** 5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2 *c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268* a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 1 26*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 546*tan(e/2 + f*x/2) **9/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2 )**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f *x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan (e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*t an(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 126*tan(e/2 + f*x/2)**8/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2...
Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (137) = 274\).
Time = 0.21 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.60 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (\frac {51 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {39 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {450 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {306 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {294 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {378 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {273 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {189 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {63 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - 19\right )}}{63 \, {\left (a^{2} c^{5} - \frac {6 \, a^{2} c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {12 \, a^{2} c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, a^{2} c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {27 \, a^{2} c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {36 \, a^{2} c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {36 \, a^{2} c^{5} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {27 \, a^{2} c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {2 \, a^{2} c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {12 \, a^{2} c^{5} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {6 \, a^{2} c^{5} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{2} c^{5} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )} f} \]
-2/63*(51*sin(f*x + e)/(cos(f*x + e) + 1) - 39*sin(f*x + e)^2/(cos(f*x + e ) + 1)^2 - 235*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 450*sin(f*x + e)^4/(c os(f*x + e) + 1)^4 - 306*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 294*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 378*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 6 3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 273*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 189*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 63*sin(f*x + e)^11/(co s(f*x + e) + 1)^11 - 19)/((a^2*c^5 - 6*a^2*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^2*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 2*a^2*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 27*a^2*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1) ^4 + 36*a^2*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 36*a^2*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 27*a^2*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^ 8 + 2*a^2*c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 12*a^2*c^5*sin(f*x + e )^10/(cos(f*x + e) + 1)^10 + 6*a^2*c^5*sin(f*x + e)^11/(cos(f*x + e) + 1)^ 11 - a^2*c^5*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)*f)
Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {\frac {21 \, {\left (21 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19\right )}}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {3591 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 19656 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 56196 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 95760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 107730 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 79464 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 38484 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10944 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1615}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2016 \, f} \]
-1/2016*(21*(21*tan(1/2*f*x + 1/2*e)^2 + 36*tan(1/2*f*x + 1/2*e) + 19)/(a^ 2*c^5*(tan(1/2*f*x + 1/2*e) + 1)^3) + (3591*tan(1/2*f*x + 1/2*e)^8 - 19656 *tan(1/2*f*x + 1/2*e)^7 + 56196*tan(1/2*f*x + 1/2*e)^6 - 95760*tan(1/2*f*x + 1/2*e)^5 + 107730*tan(1/2*f*x + 1/2*e)^4 - 79464*tan(1/2*f*x + 1/2*e)^3 + 38484*tan(1/2*f*x + 1/2*e)^2 - 10944*tan(1/2*f*x + 1/2*e) + 1615)/(a^2* c^5*(tan(1/2*f*x + 1/2*e) - 1)^9))/f
Time = 9.01 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {2\,\left (63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}-189\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+273\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-378\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+294\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+306\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-450\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+235\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+39\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-51\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+19\right )}{63\,a^2\,c^5\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^9\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \]
-(2*(39*tan(e/2 + (f*x)/2)^2 - 51*tan(e/2 + (f*x)/2) + 235*tan(e/2 + (f*x) /2)^3 - 450*tan(e/2 + (f*x)/2)^4 + 306*tan(e/2 + (f*x)/2)^5 + 294*tan(e/2 + (f*x)/2)^6 - 378*tan(e/2 + (f*x)/2)^7 + 63*tan(e/2 + (f*x)/2)^8 + 273*ta n(e/2 + (f*x)/2)^9 - 189*tan(e/2 + (f*x)/2)^10 + 63*tan(e/2 + (f*x)/2)^11 + 19))/(63*a^2*c^5*f*(tan(e/2 + (f*x)/2) - 1)^9*(tan(e/2 + (f*x)/2) + 1)^3 )