Integrand size = 32, antiderivative size = 121 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac {2 (a+a \sec (e+f x))^3 \tan (e+f x)}{99 c f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^3 \tan (e+f x)}{693 c^2 f (c-c \sec (e+f x))^4} \]
-1/11*(a+a*sec(f*x+e))^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^6-2/99*(a+a*sec(f*x +e))^3*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^5-2/693*(a+a*sec(f*x+e))^3*tan(f*x+ e)/c^2/f/(c-c*sec(f*x+e))^4
Time = 0.75 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.49 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=-\frac {a^3 (1+\sec (e+f x))^3 \left (79-18 \sec (e+f x)+2 \sec ^2(e+f x)\right ) \tan (e+f x)}{693 c^6 f (-1+\sec (e+f x))^6} \]
-1/693*(a^3*(1 + Sec[e + f*x])^3*(79 - 18*Sec[e + f*x] + 2*Sec[e + f*x]^2) *Tan[e + f*x])/(c^6*f*(-1 + Sec[e + f*x])^6)
Time = 0.60 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 4439, 3042, 4439, 3042, 4438}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^3}{(c-c \sec (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^6}dx\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {2 \int \frac {\sec (e+f x) (\sec (e+f x) a+a)^3}{(c-c \sec (e+f x))^5}dx}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5}dx}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)^3}{(c-c \sec (e+f x))^4}dx}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 4438 |
| \(\displaystyle \frac {2 \left (-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{63 c f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\) |
-1/11*((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^6) + ( 2*(-1/9*((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^5) - ((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(63*c*f*(c - c*Sec[e + f*x])^4)))/( 11*c)
3.1.32.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] /; Fre eQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] & & EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp [(m + n + 1)/(a*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* (c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0 ] && !LtQ[n, 0] && !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])
Time = 1.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42
| method | result | size |
| parallelrisch | \(-\frac {a^{3} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} \left (63 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-154 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+99\right )}{2772 c^{6} f}\) | \(51\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {2}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\right )}{4 f \,c^{6}}\) | \(52\) |
| default | \(\frac {a^{3} \left (-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {2}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\right )}{4 f \,c^{6}}\) | \(52\) |
| risch | \(\frac {2 i a^{3} \left (693 \,{\mathrm e}^{10 i \left (f x +e \right )}-1386 \,{\mathrm e}^{9 i \left (f x +e \right )}+8085 \,{\mathrm e}^{8 i \left (f x +e \right )}-10626 \,{\mathrm e}^{7 i \left (f x +e \right )}+21252 \,{\mathrm e}^{6 i \left (f x +e \right )}-15246 \,{\mathrm e}^{5 i \left (f x +e \right )}+15444 \,{\mathrm e}^{4 i \left (f x +e \right )}-4950 \,{\mathrm e}^{3 i \left (f x +e \right )}+2959 \,{\mathrm e}^{2 i \left (f x +e \right )}-176 \,{\mathrm e}^{i \left (f x +e \right )}+79\right )}{693 f \,c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{11}}\) | \(138\) |
-1/2772*a^3*cot(1/2*f*x+1/2*e)^7*(63*cot(1/2*f*x+1/2*e)^4-154*cot(1/2*f*x+ 1/2*e)^2+99)/c^6/f
Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.39 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=\frac {79 \, a^{3} \cos \left (f x + e\right )^{6} + 298 \, a^{3} \cos \left (f x + e\right )^{5} + 404 \, a^{3} \cos \left (f x + e\right )^{4} + 216 \, a^{3} \cos \left (f x + e\right )^{3} + 19 \, a^{3} \cos \left (f x + e\right )^{2} - 10 \, a^{3} \cos \left (f x + e\right ) + 2 \, a^{3}}{693 \, {\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - c^{6} f\right )} \sin \left (f x + e\right )} \]
1/693*(79*a^3*cos(f*x + e)^6 + 298*a^3*cos(f*x + e)^5 + 404*a^3*cos(f*x + e)^4 + 216*a^3*cos(f*x + e)^3 + 19*a^3*cos(f*x + e)^2 - 10*a^3*cos(f*x + e ) + 2*a^3)/((c^6*f*cos(f*x + e)^5 - 5*c^6*f*cos(f*x + e)^4 + 10*c^6*f*cos( f*x + e)^3 - 10*c^6*f*cos(f*x + e)^2 + 5*c^6*f*cos(f*x + e) - c^6*f)*sin(f *x + e))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=\frac {a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{6}} \]
a**3*(Integral(sec(e + f*x)/(sec(e + f*x)**6 - 6*sec(e + f*x)**5 + 15*sec( e + f*x)**4 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*sec(e + f*x) + 1 ), x) + Integral(3*sec(e + f*x)**2/(sec(e + f*x)**6 - 6*sec(e + f*x)**5 + 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*sec(e + f *x) + 1), x) + Integral(3*sec(e + f*x)**3/(sec(e + f*x)**6 - 6*sec(e + f*x )**5 + 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*se c(e + f*x) + 1), x) + Integral(sec(e + f*x)**4/(sec(e + f*x)**6 - 6*sec(e + f*x)**5 + 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*sec(e + f*x) + 1), x))/c**6
Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (118) = 236\).
Time = 0.22 (sec) , antiderivative size = 518, normalized size of antiderivative = 4.28 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=\frac {\frac {3 \, a^{3} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {3465 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 315\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} + \frac {9 \, a^{3} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {330 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {462 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {1155 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 105\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} + \frac {5 \, a^{3} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {693 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} - \frac {a^{3} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {3465 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + 315\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}}}{110880 \, f} \]
1/110880*(3*a^3*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 990*sin(f*x + e )^4/(cos(f*x + e) + 1)^4 - 1386*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1155 *sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 315)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11) + 9*a^3*(385*si n(f*x + e)^2/(cos(f*x + e) + 1)^2 - 330*sin(f*x + e)^4/(cos(f*x + e) + 1)^ 4 - 462*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 1155*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 105)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11) + 5*a^3*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 990*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1386*sin(f*x + e)^6/ (cos(f*x + e) + 1)^6 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 693*sin( f*x + e)^10/(cos(f*x + e) + 1)^10 - 63)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11) - a^3*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 990*sin(f*x + e )^4/(cos(f*x + e) + 1)^4 - 1386*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1155 *sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 315)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11))/f
Time = 0.42 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.47 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=-\frac {99 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 154 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 63 \, a^{3}}{2772 \, c^{6} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} \]
-1/2772*(99*a^3*tan(1/2*f*x + 1/2*e)^4 - 154*a^3*tan(1/2*f*x + 1/2*e)^2 + 63*a^3)/(c^6*f*tan(1/2*f*x + 1/2*e)^11)
Time = 13.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.55 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{18\,c^6\,f}-\frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{28\,c^6\,f}-\frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{44\,c^6\,f} \]