Integrand size = 32, antiderivative size = 163 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{231 c^2 f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3} \]
-1/11*(a+a*sec(f*x+e))^2*tan(f*x+e)/f/(c-c*sec(f*x+e))^6-1/33*(a+a*sec(f*x +e))^2*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^5-2/231*(a+a*sec(f*x+e))^2*tan(f*x+ e)/c^2/f/(c-c*sec(f*x+e))^4-2/1155*(a+a*sec(f*x+e))^2*tan(f*x+e)/f/(c^2-c^ 2*sec(f*x+e))^3
Time = 1.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.42 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=\frac {a^2 (1+\sec (e+f x))^2 \left (-152+61 \sec (e+f x)-16 \sec ^2(e+f x)+2 \sec ^3(e+f x)\right ) \tan (e+f x)}{1155 c^6 f (-1+\sec (e+f x))^6} \]
(a^2*(1 + Sec[e + f*x])^2*(-152 + 61*Sec[e + f*x] - 16*Sec[e + f*x]^2 + 2* Sec[e + f*x]^3)*Tan[e + f*x])/(1155*c^6*f*(-1 + Sec[e + f*x])^6)
Time = 0.84 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4439, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^2}{(c-c \sec (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^6}dx\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {3 \int \frac {\sec (e+f x) (\sec (e+f x) a+a)^2}{(c-c \sec (e+f x))^5}dx}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5}dx}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\sec (e+f x) (\sec (e+f x) a+a)^2}{(c-c \sec (e+f x))^4}dx}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)^2}{(c-c \sec (e+f x))^3}dx}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{7 f (c-c \sec (e+f x))^4}\right )}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}dx}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{7 f (c-c \sec (e+f x))^4}\right )}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
| \(\Big \downarrow \) 4438 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{35 c f (c-c \sec (e+f x))^3}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{7 f (c-c \sec (e+f x))^4}\right )}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}\) |
-1/11*((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^6) + ( 3*(-1/9*((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^5) + (2*(-1/7*((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^4) - ((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(35*c*f*(c - c*Sec[e + f*x])^3))) /(9*c)))/(11*c)
3.1.20.3.1 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 = 64, normalized size of antiderivative = 0.39
| method | result | size |
| parallelrisch | \(-\frac {a^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \left (105 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-385 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+495 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-231\right )}{9240 c^{6} f}\) | \(64\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {3}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}\right )}{8 f \,c^{6}}\) | \(65\) |
| default | \(\frac {a^{2} \left (\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {3}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}\right )}{8 f \,c^{6}}\) | \(65\) |
| risch | \(\frac {2 i a^{2} \left (1155 \,{\mathrm e}^{10 i \left (f x +e \right )}-3465 \,{\mathrm e}^{9 i \left (f x +e \right )}+13860 \,{\mathrm e}^{8 i \left (f x +e \right )}-23100 \,{\mathrm e}^{7 i \left (f x +e \right )}+37422 \,{\mathrm e}^{6 i \left (f x +e \right )}-32802 \,{\mathrm e}^{5 i \left (f x +e \right )}+27060 \,{\mathrm e}^{4 i \left (f x +e \right )}-11220 \,{\mathrm e}^{3 i \left (f x +e \right )}+4895 \,{\mathrm e}^{2 i \left (f x +e \right )}-517 \,{\mathrm e}^{i \left (f x +e \right )}+152\right )}{1155 f \,c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{11}}\) | \(138\) |
| norman | \(\frac {-\frac {a^{2}}{88 c f}+\frac {17 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{264 c f}-\frac {137 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{924 c f}+\frac {73 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{420 c f}-\frac {29 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{280 c f}+\frac {a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{40 c f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}\) | \(153\) |
-1/9240*a^2*cot(1/2*f*x+1/2*e)^5*(105*cot(1/2*f*x+1/2*e)^6-385*cot(1/2*f*x +1/2*e)^4+495*cot(1/2*f*x+1/2*e)^2-231)/c^6/f
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=\frac {152 \, a^{2} \cos \left (f x + e\right )^{6} + 395 \, a^{2} \cos \left (f x + e\right )^{5} + 289 \, a^{2} \cos \left (f x + e\right )^{4} + 15 \, a^{2} \cos \left (f x + e\right )^{3} - 19 \, a^{2} \cos \left (f x + e\right )^{2} + 10 \, a^{2} \cos \left (f x + e\right ) - 2 \, a^{2}}{1155 \, {\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - c^{6} f\right )} \sin \left (f x + e\right )} \]
1/1155*(152*a^2*cos(f*x + e)^6 + 395*a^2*cos(f*x + e)^5 + 289*a^2*cos(f*x + e)^4 + 15*a^2*cos(f*x + e)^3 - 19*a^2*cos(f*x + e)^2 + 10*a^2*cos(f*x + e) - 2*a^2)/((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))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=\frac {a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{6}} \]
a**2*(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(2*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(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*sec( e + f*x) + 1), x))/c**6
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (159) = 318\).
Time = 0.22 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.39 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=\frac {\frac {a^{2} {\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 {6 \, a^{2} {\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^{2} {\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}}}{110880 \, f} \]
1/110880*(a^2*(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*s in(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) + 6*a^2*(385*sin( 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^2*(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/(c os(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))/f
Time = 0.41 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.45 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=\frac {231 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 495 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 385 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105 \, a^{2}}{9240 \, c^{6} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} \]
1/9240*(231*a^2*tan(1/2*f*x + 1/2*e)^6 - 495*a^2*tan(1/2*f*x + 1/2*e)^4 + 385*a^2*tan(1/2*f*x + 1/2*e)^2 - 105*a^2)/(c^6*f*tan(1/2*f*x + 1/2*e)^11)
Time = 14.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.66 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx=-\frac {a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (105\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-385\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+495\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-231\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{9240\,c^6\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]