Integrand size = 26, antiderivative size = 97 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\frac {\sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {6 \tan (e+f x)}{7 a^3 c^4 f}+\frac {4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac {6 \tan ^5(e+f x)}{35 a^3 c^4 f} \]
1/7*sec(f*x+e)^5/a^3/f/(c^4-c^4*sin(f*x+e))+6/7*tan(f*x+e)/a^3/c^4/f+4/7*t an(f*x+e)^3/a^3/c^4/f+6/35*tan(f*x+e)^5/a^3/c^4/f
Time = 1.92 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.00 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (197540 \cos (e+f x)+20480 \cos (2 (e+f x))+98770 \cos (3 (e+f x))+16384 \cos (4 (e+f x))+19754 \cos (5 (e+f x))+4096 \cos (6 (e+f x))+81920 \sin (e+f x)-49385 \sin (2 (e+f x))+40960 \sin (3 (e+f x))-39508 \sin (4 (e+f x))+8192 \sin (5 (e+f x))-9877 \sin (6 (e+f x)))}{286720 a^3 c^4 f (-1+\sin (e+f x))^4 (1+\sin (e+f x))^3} \]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])*(197540*Cos[e + f*x] + 20480*Cos[2*(e + f*x)] + 98770*Cos[3*(e + f*x)] + 16384*Cos[4*(e + f*x)] + 19754*Cos[5*(e + f*x)] + 4096*Cos[6*(e + f*x)] + 81920*Sin[e + f*x] - 49385*Sin[2*(e + f*x)] + 40960*Sin[3*(e + f*x)] - 3 9508*Sin[4*(e + f*x)] + 8192*Sin[5*(e + f*x)] - 9877*Sin[6*(e + f*x)]))/(2 86720*a^3*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3)
Time = 0.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\cos (e+f x)^6 (c-c \sin (e+f x))}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {6 \int \sec ^6(e+f x)dx}{7 c}+\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {6 \int \csc \left (e+f x+\frac {\pi }{2}\right )^6dx}{7 c}+\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}}{a^3 c^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {6 \int \left (\tan ^4(e+f x)+2 \tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{7 c f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {6 \left (-\frac {1}{5} \tan ^5(e+f x)-\frac {2}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{7 c f}}{a^3 c^3}\) |
(Sec[e + f*x]^5/(7*f*(c - c*Sin[e + f*x])) - (6*(-Tan[e + f*x] - (2*Tan[e + f*x]^3)/3 - Tan[e + f*x]^5/5))/(7*c*f))/(a^3*c^3)
3.3.87.3.1 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 = 2.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {-\frac {32 i}{35}+\frac {64 \,{\mathrm e}^{i \left (f x +e \right )}}{35}+\frac {128 \,{\mathrm e}^{5 i \left (f x +e \right )}}{7}+\frac {64 \,{\mathrm e}^{3 i \left (f x +e \right )}}{7}-\frac {128 i {\mathrm e}^{2 i \left (f x +e \right )}}{35}-\frac {32 i {\mathrm e}^{4 i \left (f x +e \right )}}{7}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,a^{3} c^{4}}\) | \(100\) |
| parallelrisch | \(\frac {-\frac {2}{7}+2 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {52 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}-2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-6 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {52 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {52 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {36 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {6 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7}+\frac {22 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}}{f \,a^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(181\) |
| derivativedivides | \(\frac {-\frac {2}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {11}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {11}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {21}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {11}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{3} c^{4} f}\) | \(193\) |
| default | \(\frac {-\frac {2}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {11}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {11}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {21}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {11}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{3} c^{4} f}\) | \(193\) |
| norman | \(\frac {-\frac {6 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {52 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}-\frac {2}{7 a c f}-\frac {2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {2 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {2 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 a c f}+\frac {22 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {36 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}-\frac {6 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {52 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {52 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 a c f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(286\) |
32/35*(-I+2*exp(I*(f*x+e))+20*exp(5*I*(f*x+e))+10*exp(3*I*(f*x+e))-4*I*exp (2*I*(f*x+e))-5*I*exp(4*I*(f*x+e)))/(exp(I*(f*x+e))-I)^7/(exp(I*(f*x+e))+I )^5/f/a^3/c^4
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=-\frac {16 \, \cos \left (f x + e\right )^{6} - 8 \, \cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 2 \, {\left (8 \, \cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{2} + 3\right )} \sin \left (f x + e\right ) - 1}{35 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - a^{3} c^{4} f \cos \left (f x + e\right )^{5}\right )}} \]
-1/35*(16*cos(f*x + e)^6 - 8*cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 2*(8*cos( f*x + e)^4 + 4*cos(f*x + e)^2 + 3)*sin(f*x + e) - 1)/(a^3*c^4*f*cos(f*x + e)^5*sin(f*x + e) - a^3*c^4*f*cos(f*x + e)^5)
Leaf count of result is larger than twice the leaf count of optimal. 3186 vs. \(2 (87) = 174\).
Time = 20.21 (sec) , antiderivative size = 3186, normalized size of antiderivative = 32.85 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
Piecewise((-70*tan(e/2 + f*x/2)**11/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**1 0 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2) **8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/ 2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f* x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f *x/2) - 35*a**3*c**4*f) + 70*tan(e/2 + f*x/2)**10/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e /2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*ta n(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f* tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4* f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4 *f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) + 70*tan(e/2 + f*x/2)**9/(35*a**3*c* *4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3 *c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a **3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700 *a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 3 50*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) - 210*tan(e/2 + f*x/2)* *8/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/...
Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (90) = 180\).
Time = 0.22 (sec) , antiderivative size = 519, normalized size of antiderivative = 5.35 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\frac {2 \, {\left (\frac {25 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {55 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {130 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {26 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {182 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {126 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {105 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {35 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {35 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {35 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + 5\right )}}{35 \, {\left (a^{3} c^{4} - \frac {2 \, a^{3} c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {4 \, a^{3} c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {20 \, a^{3} c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {20 \, a^{3} c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {5 \, a^{3} c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {10 \, a^{3} c^{4} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {4 \, a^{3} c^{4} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {2 \, a^{3} c^{4} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{3} c^{4} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )} f} \]
2/35*(25*sin(f*x + e)/(cos(f*x + e) + 1) - 55*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 130*sin(f*x + e)^4/(cos (f*x + e) + 1)^4 + 26*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 182*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 126*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 105* sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 35*sin(f*x + e)^9/(cos(f*x + e) + 1) ^9 - 35*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 35*sin(f*x + e)^11/(cos(f* x + e) + 1)^11 + 5)/((a^3*c^4 - 2*a^3*c^4*sin(f*x + e)/(cos(f*x + e) + 1) - 4*a^3*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*c^4*sin(f*x + e)^ 3/(cos(f*x + e) + 1)^3 + 5*a^3*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 2 0*a^3*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*c^4*sin(f*x + e)^7/ (cos(f*x + e) + 1)^7 - 5*a^3*c^4*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 10* a^3*c^4*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 4*a^3*c^4*sin(f*x + e)^10/(c os(f*x + e) + 1)^10 + 2*a^3*c^4*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - a^ 3*c^4*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)*f)
Time = 0.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=-\frac {\frac {7 \, {\left (55 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 180 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 250 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 160 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 43\right )}}{a^{3} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {735 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3360 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 7315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 8820 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6321 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2492 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 461}{a^{3} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{560 \, f} \]
-1/560*(7*(55*tan(1/2*f*x + 1/2*e)^4 + 180*tan(1/2*f*x + 1/2*e)^3 + 250*ta n(1/2*f*x + 1/2*e)^2 + 160*tan(1/2*f*x + 1/2*e) + 43)/(a^3*c^4*(tan(1/2*f* x + 1/2*e) + 1)^5) + (735*tan(1/2*f*x + 1/2*e)^6 - 3360*tan(1/2*f*x + 1/2* e)^5 + 7315*tan(1/2*f*x + 1/2*e)^4 - 8820*tan(1/2*f*x + 1/2*e)^3 + 6321*ta n(1/2*f*x + 1/2*e)^2 - 2492*tan(1/2*f*x + 1/2*e) + 461)/(a^3*c^4*(tan(1/2* f*x + 1/2*e) - 1)^7))/f
Time = 9.46 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=-\frac {2\,\left (35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+105\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-182\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+26\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+130\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-55\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+25\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+5\right )}{35\,a^3\,c^4\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^7\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5} \]
-(2*(25*tan(e/2 + (f*x)/2) - 55*tan(e/2 + (f*x)/2)^2 + 15*tan(e/2 + (f*x)/ 2)^3 + 130*tan(e/2 + (f*x)/2)^4 + 26*tan(e/2 + (f*x)/2)^5 - 182*tan(e/2 + (f*x)/2)^6 + 126*tan(e/2 + (f*x)/2)^7 + 105*tan(e/2 + (f*x)/2)^8 - 35*tan( e/2 + (f*x)/2)^9 - 35*tan(e/2 + (f*x)/2)^10 + 35*tan(e/2 + (f*x)/2)^11 + 5 ))/(35*a^3*c^4*f*(tan(e/2 + (f*x)/2) - 1)^7*(tan(e/2 + (f*x)/2) + 1)^5)