Integrand size = 32, antiderivative size = 80 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f} \]
1/5*cot(f*x+e)^5/a^2/c^3/f+csc(f*x+e)/a^2/c^3/f-2/3*csc(f*x+e)^3/a^2/c^3/f +1/5*csc(f*x+e)^5/a^2/c^3/f
Time = 0.83 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\left (3+12 \sec (e+f x)-12 \sec ^2(e+f x)-8 \sec ^3(e+f x)+8 \sec ^4(e+f x)\right ) \tan (e+f x)}{15 a^2 c^3 f (-1+\sec (e+f x))^3 (1+\sec (e+f x))^2} \]
((3 + 12*Sec[e + f*x] - 12*Sec[e + f*x]^2 - 8*Sec[e + f*x]^3 + 8*Sec[e + f *x]^4)*Tan[e + f*x])/(15*a^2*c^3*f*(-1 + Sec[e + f*x])^3*(1 + Sec[e + f*x] )^2)
Time = 0.36 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86, 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.
\(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4446 |
\(\displaystyle -\frac {\int \left (a \csc (e+f x) \cot ^5(e+f x)+a \csc ^2(e+f x) \cot ^4(e+f x)\right )dx}{a^3 c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {a \cot ^5(e+f x)}{5 f}-\frac {a \csc ^5(e+f x)}{5 f}+\frac {2 a \csc ^3(e+f x)}{3 f}-\frac {a \csc (e+f x)}{f}}{a^3 c^3}\) |
-((-1/5*(a*Cot[e + f*x]^5)/f - (a*Csc[e + f*x])/f + (2*a*Csc[e + f*x]^3)/( 3*f) - (a*Csc[e + f*x]^5)/(5*f))/(a^3*c^3))
3.1.49.3.1 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 = 0.66 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {3 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-20 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+60 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+90 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{240 f \,a^{2} c^{3}}\) | \(74\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}+\frac {6}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{16 f \,a^{2} c^{3}}\) | \(76\) |
default | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}+\frac {6}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{16 f \,a^{2} c^{3}}\) | \(76\) |
risch | \(\frac {2 i \left (15 \,{\mathrm e}^{7 i \left (f x +e \right )}-15 \,{\mathrm e}^{6 i \left (f x +e \right )}-5 \,{\mathrm e}^{5 i \left (f x +e \right )}+25 \,{\mathrm e}^{4 i \left (f x +e \right )}+13 \,{\mathrm e}^{3 i \left (f x +e \right )}-21 \,{\mathrm e}^{2 i \left (f x +e \right )}+9 \,{\mathrm e}^{i \left (f x +e \right )}+3\right )}{15 f \,a^{2} c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\) | \(118\) |
norman | \(\frac {\frac {1}{80 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{12 a c f}+\frac {3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{8 a c f}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{4 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{48 a c f}}{a \,c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(119\) |
1/240*(3*cot(1/2*f*x+1/2*e)^5-5*tan(1/2*f*x+1/2*e)^3-20*cot(1/2*f*x+1/2*e) ^3+60*tan(1/2*f*x+1/2*e)+90*cot(1/2*f*x+1/2*e))/f/a^2/c^3
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {3 \, \cos \left (f x + e\right )^{4} + 12 \, \cos \left (f x + e\right )^{3} - 12 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 8}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )} \]
1/15*(3*cos(f*x + e)^4 + 12*cos(f*x + e)^3 - 12*cos(f*x + e)^2 - 8*cos(f*x + e) + 8)/((a^2*c^3*f*cos(f*x + e)^3 - a^2*c^3*f*cos(f*x + e)^2 - a^2*c^3 *f*cos(f*x + e) + a^2*c^3*f)*sin(f*x + e))
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=- \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c^{3}} \]
-Integral(sec(e + f*x)/(sec(e + f*x)**5 - sec(e + f*x)**4 - 2*sec(e + f*x) **3 + 2*sec(e + f*x)**2 + sec(e + f*x) - 1), x)/(a**2*c**3)
Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.51 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\frac {5 \, {\left (\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{3}} - \frac {{\left (\frac {20 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {90 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{a^{2} c^{3} \sin \left (f x + e\right )^{5}}}{240 \, f} \]
1/240*(5*(12*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e ) + 1)^3)/(a^2*c^3) - (20*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 90*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3)*(cos(f*x + e) + 1)^5/(a^2*c^3*sin(f*x + e)^5))/f
Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\frac {90 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3}{a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} - \frac {5 \, {\left (a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{9}}}{240 \, f} \]
1/240*((90*tan(1/2*f*x + 1/2*e)^4 - 20*tan(1/2*f*x + 1/2*e)^2 + 3)/(a^2*c^ 3*tan(1/2*f*x + 1/2*e)^5) - 5*(a^4*c^6*tan(1/2*f*x + 1/2*e)^3 - 12*a^4*c^6 *tan(1/2*f*x + 1/2*e))/(a^6*c^9))/f
Time = 13.47 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+60\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3}{240\,a^2\,c^3\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]