Integrand size = 26, antiderivative size = 111 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{7 a^2 c^4 f}+\frac {4 \tan ^3(e+f x)}{21 a^2 c^4 f} \] Output:
1/7*sec(f*x+e)^3/a^2/f/(c^2-c^2*sin(f*x+e))^2+1/7*sec(f*x+e)^3/a^2/f/(c^4- c^4*sin(f*x+e))+4/7*tan(f*x+e)/a^2/c^4/f+4/21*tan(f*x+e)^3/a^2/c^4/f
Time = 1.94 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (54390 \cos (e+f x)+8192 \cos (2 (e+f x))+11655 \cos (3 (e+f x))+4096 \cos (4 (e+f x))-3885 \cos (5 (e+f x))+14336 \sin (e+f x)-31080 \sin (2 (e+f x))+3072 \sin (3 (e+f x))-15540 \sin (4 (e+f x))-1024 \sin (5 (e+f x)))}{43008 a^2 c^4 f (-1+\sin (e+f x))^4 (1+\sin (e+f x))^2} \] Input:
Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4),x]
Output:
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])*(54390*Cos[e + f*x] + 8192*Cos[2*(e + f*x)] + 11655*Cos[3*(e + f*x)] + 4096*Cos[4*(e + f*x)] - 3885*Cos[5*(e + f*x)] + 14336*Sin[e + f*x] - 31080 *Sin[2*(e + f*x)] + 3072*Sin[3*(e + f*x)] - 15540*Sin[4*(e + f*x)] - 1024* Sin[5*(e + f*x)]))/(43008*a^2*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x ])^2)
Time = 0.53 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3042, 3215, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^4}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))^2}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {5 \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)}dx}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))}dx}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {5 \left (\frac {4 \int \sec ^4(e+f x)dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {4 \int \csc \left (e+f x+\frac {\pi }{2}\right )^4dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {5 \left (\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \int \left (\tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{5 c f}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}+\frac {5 \left (\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \left (-\frac {1}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{5 c f}\right )}{7 c}}{a^2 c^2}\) |
Input:
Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4),x]
Output:
(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 ))/(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 = 1.65 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(-\frac {16 i \left (-8 i {\mathrm e}^{3 i \left (f x +e \right )}+14 \,{\mathrm e}^{4 i \left (f x +e \right )}-4 i {\mathrm e}^{i \left (f x +e \right )}+3 \,{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{21 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,a^{2} c^{4}}\) | \(89\) |
| parallelrisch | \(\frac {-\frac {4}{7}-\frac {152 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{21}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3}-\frac {16 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7}+\frac {16 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{7}}{f \,a^{2} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(155\) |
| derivativedivides | \(\frac {-\frac {4}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {55}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {23}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {13}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} c^{4} f}\) | \(163\) |
| default | \(\frac {-\frac {4}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {55}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {23}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {13}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} c^{4} f}\) | \(163\) |
| norman | \(\frac {\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{a f c}-\frac {1}{14 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{2 a f c}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{2 a f c}-\frac {20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 a f c}-\frac {12 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 a c f}-\frac {68 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{21 a c f}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3 a f c}-\frac {13 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 a f c}+\frac {53 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{14 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(242\) |
Input:
int(1/(a+sin(f*x+e)*a)^2/(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
-16/21*I*(-8*I*exp(3*I*(f*x+e))+14*exp(4*I*(f*x+e))-4*I*exp(I*(f*x+e))+3*e xp(2*I*(f*x+e))-1)/(exp(I*(f*x+e))-I)^7/(exp(I*(f*x+e))+I)^3/f/a^2/c^4
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=-\frac {16 \, \cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{2} - {\left (8 \, \cos \left (f x + e\right )^{4} - 12 \, \cos \left (f x + e\right )^{2} - 5\right )} \sin \left (f x + e\right ) - 2}{21 \, {\left (a^{2} c^{4} f \cos \left (f x + e\right )^{5} + 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3}\right )}} \] Input:
integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm="fricas")
Output:
-1/21*(16*cos(f*x + e)^4 - 8*cos(f*x + e)^2 - (8*cos(f*x + e)^4 - 12*cos(f *x + e)^2 - 5)*sin(f*x + e) - 2)/(a^2*c^4*f*cos(f*x + e)^5 + 2*a^2*c^4*f*c os(f*x + e)^3*sin(f*x + e) - 2*a^2*c^4*f*cos(f*x + e)^3)
Leaf count of result is larger than twice the leaf count of optimal. 2213 vs. \(2 (97) = 194\).
Time = 10.29 (sec) , antiderivative size = 2213, normalized size of antiderivative = 19.94 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**4,x)
Output:
Piecewise((-42*tan(e/2 + f*x/2)**9/(21*a**2*c**4*f*tan(e/2 + f*x/2)**10 - 84*a**2*c**4*f*tan(e/2 + f*x/2)**9 + 63*a**2*c**4*f*tan(e/2 + f*x/2)**8 + 168*a**2*c**4*f*tan(e/2 + f*x/2)**7 - 294*a**2*c**4*f*tan(e/2 + f*x/2)**6 + 294*a**2*c**4*f*tan(e/2 + f*x/2)**4 - 168*a**2*c**4*f*tan(e/2 + f*x/2)** 3 - 63*a**2*c**4*f*tan(e/2 + f*x/2)**2 + 84*a**2*c**4*f*tan(e/2 + f*x/2) - 21*a**2*c**4*f) + 84*tan(e/2 + f*x/2)**8/(21*a**2*c**4*f*tan(e/2 + f*x/2) **10 - 84*a**2*c**4*f*tan(e/2 + f*x/2)**9 + 63*a**2*c**4*f*tan(e/2 + f*x/2 )**8 + 168*a**2*c**4*f*tan(e/2 + f*x/2)**7 - 294*a**2*c**4*f*tan(e/2 + f*x /2)**6 + 294*a**2*c**4*f*tan(e/2 + f*x/2)**4 - 168*a**2*c**4*f*tan(e/2 + f *x/2)**3 - 63*a**2*c**4*f*tan(e/2 + f*x/2)**2 + 84*a**2*c**4*f*tan(e/2 + f *x/2) - 21*a**2*c**4*f) - 56*tan(e/2 + f*x/2)**7/(21*a**2*c**4*f*tan(e/2 + f*x/2)**10 - 84*a**2*c**4*f*tan(e/2 + f*x/2)**9 + 63*a**2*c**4*f*tan(e/2 + f*x/2)**8 + 168*a**2*c**4*f*tan(e/2 + f*x/2)**7 - 294*a**2*c**4*f*tan(e/ 2 + f*x/2)**6 + 294*a**2*c**4*f*tan(e/2 + f*x/2)**4 - 168*a**2*c**4*f*tan( e/2 + f*x/2)**3 - 63*a**2*c**4*f*tan(e/2 + f*x/2)**2 + 84*a**2*c**4*f*tan( e/2 + f*x/2) - 21*a**2*c**4*f) - 112*tan(e/2 + f*x/2)**6/(21*a**2*c**4*f*t an(e/2 + f*x/2)**10 - 84*a**2*c**4*f*tan(e/2 + f*x/2)**9 + 63*a**2*c**4*f* tan(e/2 + f*x/2)**8 + 168*a**2*c**4*f*tan(e/2 + f*x/2)**7 - 294*a**2*c**4* f*tan(e/2 + f*x/2)**6 + 294*a**2*c**4*f*tan(e/2 + f*x/2)**4 - 168*a**2*c** 4*f*tan(e/2 + f*x/2)**3 - 63*a**2*c**4*f*tan(e/2 + f*x/2)**2 + 84*a**2*...
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (105) = 210\).
Time = 0.05 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.85 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {24 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {76 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {28 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {42 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {56 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {28 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {42 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {21 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - 6\right )}}{21 \, {\left (a^{2} c^{4} - \frac {4 \, a^{2} c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {8 \, a^{2} c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {8 \, a^{2} c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {4 \, a^{2} c^{4} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {a^{2} c^{4} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}}\right )} f} \] Input:
integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm="maxima")
Output:
-2/21*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 24*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 76*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 28*sin(f*x + e)^4/(cos( f*x + e) + 1)^4 + 42*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 56*sin(f*x + e) ^6/(cos(f*x + e) + 1)^6 - 28*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 42*sin( f*x + e)^8/(cos(f*x + e) + 1)^8 - 21*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 6)/((a^2*c^4 - 4*a^2*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*c^4*sin( f*x + e)^2/(cos(f*x + e) + 1)^2 + 8*a^2*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 14*a^2*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 14*a^2*c^4*sin(f* x + e)^6/(cos(f*x + e) + 1)^6 - 8*a^2*c^4*sin(f*x + e)^7/(cos(f*x + e) + 1 )^7 - 3*a^2*c^4*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 4*a^2*c^4*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - a^2*c^4*sin(f*x + e)^10/(cos(f*x + e) + 1)^10) *f)
Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=-\frac {\frac {7 \, {\left (9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8\right )}}{a^{2} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {273 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1155 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2450 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2870 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2037 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 791 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 152}{a^{2} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{168 \, f} \] Input:
integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm="giac")
Output:
-1/168*(7*(9*tan(1/2*f*x + 1/2*e)^2 + 15*tan(1/2*f*x + 1/2*e) + 8)/(a^2*c^ 4*(tan(1/2*f*x + 1/2*e) + 1)^3) + (273*tan(1/2*f*x + 1/2*e)^6 - 1155*tan(1 /2*f*x + 1/2*e)^5 + 2450*tan(1/2*f*x + 1/2*e)^4 - 2870*tan(1/2*f*x + 1/2*e )^3 + 2037*tan(1/2*f*x + 1/2*e)^2 - 791*tan(1/2*f*x + 1/2*e) + 152)/(a^2*c ^4*(tan(1/2*f*x + 1/2*e) - 1)^7))/f
Time = 17.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=-\frac {\frac {\sin \left (e+f\,x\right )}{3}+\frac {4\,\cos \left (2\,e+2\,f\,x\right )}{21}+\frac {2\,\cos \left (4\,e+4\,f\,x\right )}{21}+\frac {\sin \left (3\,e+3\,f\,x\right )}{14}-\frac {\sin \left (5\,e+5\,f\,x\right )}{42}}{a^2\,c^4\,f\,\left (\frac {\cos \left (5\,e+5\,f\,x\right )}{16}-\frac {3\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {7\,\cos \left (e+f\,x\right )}{8}+\frac {\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {\sin \left (4\,e+4\,f\,x\right )}{4}\right )} \] Input:
int(1/((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^4),x)
Output:
-(sin(e + f*x)/3 + (4*cos(2*e + 2*f*x))/21 + (2*cos(4*e + 4*f*x))/21 + sin (3*e + 3*f*x)/14 - sin(5*e + 5*f*x)/42)/(a^2*c^4*f*(cos(5*e + 5*f*x)/16 - (3*cos(3*e + 3*f*x))/16 - (7*cos(e + f*x))/8 + sin(2*e + 2*f*x)/2 + sin(4* e + 4*f*x)/4))
Time = 0.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx=\frac {-9 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+18 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-18 \cos \left (f x +e \right ) \sin \left (f x +e \right )+9 \cos \left (f x +e \right )+16 \sin \left (f x +e \right )^{5}-32 \sin \left (f x +e \right )^{4}-8 \sin \left (f x +e \right )^{3}+48 \sin \left (f x +e \right )^{2}-18 \sin \left (f x +e \right )-12}{42 \cos \left (f x +e \right ) a^{2} c^{4} f \left (\sin \left (f x +e \right )^{4}-2 \sin \left (f x +e \right )^{3}+2 \sin \left (f x +e \right )-1\right )} \] Input:
int(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x)
Output:
( - 9*cos(e + f*x)*sin(e + f*x)**4 + 18*cos(e + f*x)*sin(e + f*x)**3 - 18* cos(e + f*x)*sin(e + f*x) + 9*cos(e + f*x) + 16*sin(e + f*x)**5 - 32*sin(e + f*x)**4 - 8*sin(e + f*x)**3 + 48*sin(e + f*x)**2 - 18*sin(e + f*x) - 12 )/(42*cos(e + f*x)*a**2*c**4*f*(sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*si n(e + f*x) - 1))