Integrand size = 32, antiderivative size = 99 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=-\frac {\cot ^7(e+f x)}{7 a^3 c^4 f}+\frac {\csc (e+f x)}{a^3 c^4 f}-\frac {\csc ^3(e+f x)}{a^3 c^4 f}+\frac {3 \csc ^5(e+f x)}{5 a^3 c^4 f}-\frac {\csc ^7(e+f x)}{7 a^3 c^4 f} \]
-1/7*cot(f*x+e)^7/a^3/c^4/f+csc(f*x+e)/a^3/c^4/f-csc(f*x+e)^3/a^3/c^4/f+3/ 5*csc(f*x+e)^5/a^3/c^4/f-1/7*csc(f*x+e)^7/a^3/c^4/f
Time = 3.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\left (-5-30 \sec (e+f x)+30 \sec ^2(e+f x)+40 \sec ^3(e+f x)-40 \sec ^4(e+f x)-16 \sec ^5(e+f x)+16 \sec ^6(e+f x)\right ) \tan (e+f x)}{35 a^3 c^4 f (-1+\sec (e+f x))^4 (1+\sec (e+f x))^3} \]
((-5 - 30*Sec[e + f*x] + 30*Sec[e + f*x]^2 + 40*Sec[e + f*x]^3 - 40*Sec[e + f*x]^4 - 16*Sec[e + f*x]^5 + 16*Sec[e + f*x]^6)*Tan[e + f*x])/(35*a^3*c^ 4*f*(-1 + Sec[e + f*x])^4*(1 + Sec[e + f*x])^3)
Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3042, 4446, 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.
| \(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4446 |
| \(\displaystyle \frac {\int \left (a \csc (e+f x) \cot ^7(e+f x)+a \csc ^2(e+f x) \cot ^6(e+f x)\right )dx}{a^4 c^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a \cot ^7(e+f x)}{7 f}-\frac {a \csc ^7(e+f x)}{7 f}+\frac {3 a \csc ^5(e+f x)}{5 f}-\frac {a \csc ^3(e+f x)}{f}+\frac {a \csc (e+f x)}{f}}{a^4 c^4}\) |
(-1/7*(a*Cot[e + f*x]^7)/f + (a*Csc[e + f*x])/f - (a*Csc[e + f*x]^3)/f + ( 3*a*Csc[e + f*x]^5)/(5*f) - (a*Csc[e + f*x]^7)/(7*f))/(a^4*c^4)
3.1.61.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m ), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && Eq Q[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
Time = 0.79 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(-\frac {\left (-90 \cos \left (4 f x +4 e \right )+5 \cos \left (6 f x +6 e \right )+152 \cos \left (f x +e \right )+60 \cos \left (5 f x +5 e \right )-182-20 \cos \left (3 f x +3 e \right )+235 \cos \left (2 f x +2 e \right )\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{71680 f \,a^{3} c^{4}}\) | \(99\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+15 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {20}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {6}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {5}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}}{64 f \,c^{4} a^{3}}\) | \(102\) |
| default | \(\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+15 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {20}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {6}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {5}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}}{64 f \,c^{4} a^{3}}\) | \(102\) |
| risch | \(\frac {2 i \left (35 \,{\mathrm e}^{11 i \left (f x +e \right )}-35 \,{\mathrm e}^{10 i \left (f x +e \right )}-35 \,{\mathrm e}^{9 i \left (f x +e \right )}+105 \,{\mathrm e}^{8 i \left (f x +e \right )}+126 \,{\mathrm e}^{7 i \left (f x +e \right )}-182 \,{\mathrm e}^{6 i \left (f x +e \right )}+26 \,{\mathrm e}^{5 i \left (f x +e \right )}+130 \,{\mathrm e}^{4 i \left (f x +e \right )}+15 \,{\mathrm e}^{3 i \left (f x +e \right )}-55 \,{\mathrm e}^{2 i \left (f x +e \right )}+25 \,{\mathrm e}^{i \left (f x +e \right )}+5\right )}{35 f \,c^{4} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7}}\) | \(162\) |
| norman | \(\frac {-\frac {1}{448 a c f}+\frac {3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{160 a c f}-\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{64 a c f}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{16 a c f}+\frac {15 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{64 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{32 a c f}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{320 a c f}}{a^{2} c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\) | \(163\) |
-1/71680*(-90*cos(4*f*x+4*e)+5*cos(6*f*x+6*e)+152*cos(f*x+e)+60*cos(5*f*x+ 5*e)-182-20*cos(3*f*x+3*e)+235*cos(2*f*x+2*e))*sec(1/2*f*x+1/2*e)^5*csc(1/ 2*f*x+1/2*e)^7/f/a^3/c^4
Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.65 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {5 \, \cos \left (f x + e\right )^{6} + 30 \, \cos \left (f x + e\right )^{5} - 30 \, \cos \left (f x + e\right )^{4} - 40 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} + 16 \, \cos \left (f x + e\right ) - 16}{35 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{5} - a^{3} c^{4} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{4} f \cos \left (f x + e\right )^{3} + 2 \, a^{3} c^{4} f \cos \left (f x + e\right )^{2} + a^{3} c^{4} f \cos \left (f x + e\right ) - a^{3} c^{4} f\right )} \sin \left (f x + e\right )} \]
1/35*(5*cos(f*x + e)^6 + 30*cos(f*x + e)^5 - 30*cos(f*x + e)^4 - 40*cos(f* x + e)^3 + 40*cos(f*x + e)^2 + 16*cos(f*x + e) - 16)/((a^3*c^4*f*cos(f*x + e)^5 - a^3*c^4*f*cos(f*x + e)^4 - 2*a^3*c^4*f*cos(f*x + e)^3 + 2*a^3*c^4* f*cos(f*x + e)^2 + a^3*c^4*f*cos(f*x + e) - a^3*c^4*f)*sin(f*x + e))
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - \sec ^{6}{\left (e + f x \right )} - 3 \sec ^{5}{\left (e + f x \right )} + 3 \sec ^{4}{\left (e + f x \right )} + 3 \sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} - \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{4}} \]
Integral(sec(e + f*x)/(sec(e + f*x)**7 - sec(e + f*x)**6 - 3*sec(e + f*x)* *5 + 3*sec(e + f*x)**4 + 3*sec(e + f*x)**3 - 3*sec(e + f*x)**2 - sec(e + f *x) + 1), x)/(a**3*c**4)
Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\frac {7 \, {\left (\frac {75 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{4}} + \frac {{\left (\frac {42 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {175 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {700 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{a^{3} c^{4} \sin \left (f x + e\right )^{7}}}{2240 \, f} \]
1/2240*(7*(75*sin(f*x + e)/(cos(f*x + e) + 1) - 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/(a^3*c^4) + (42*sin(f* x + e)^2/(cos(f*x + e) + 1)^2 - 175*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 700*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 5)*(cos(f*x + e) + 1)^7/(a^3*c^4 *sin(f*x + e)^7))/f
Time = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.29 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\frac {700 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 175 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 42 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5}{a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} + \frac {7 \, {\left (a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 75 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{20}}}{2240 \, f} \]
1/2240*((700*tan(1/2*f*x + 1/2*e)^6 - 175*tan(1/2*f*x + 1/2*e)^4 + 42*tan( 1/2*f*x + 1/2*e)^2 - 5)/(a^3*c^4*tan(1/2*f*x + 1/2*e)^7) + 7*(a^12*c^16*ta n(1/2*f*x + 1/2*e)^5 - 10*a^12*c^16*tan(1/2*f*x + 1/2*e)^3 + 75*a^12*c^16* tan(1/2*f*x + 1/2*e))/(a^15*c^20))/f
Time = 14.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.30 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\left (2\,{\sin \left (\frac {e}{4}+\frac {f\,x}{4}\right )}^2-1\right )\,\left (\frac {235\,{\sin \left (e+f\,x\right )}^2}{16}-\frac {45\,{\sin \left (2\,e+2\,f\,x\right )}^2}{8}+\frac {19\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{2}+\frac {5\,{\sin \left (3\,e+3\,f\,x\right )}^2}{16}-\frac {5\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2}{4}+\frac {15\,{\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}^2}{4}-5\right )}{2240\,a^3\,c^4\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,{\left ({\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^3} \]