Integrand size = 30, antiderivative size = 158 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{21 c f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{105 c^2 f (c-c \sec (e+f x))^3}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{315 c f \left (c^2-c^2 \sec (e+f x)\right )^2} \]
-1/9*(a+a*sec(f*x+e))*tan(f*x+e)/f/(c-c*sec(f*x+e))^5-1/21*(a+a*sec(f*x+e) )*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^4-2/105*(a+a*sec(f*x+e))*tan(f*x+e)/c^2/ f/(c-c*sec(f*x+e))^3-2/315*(a+a*sec(f*x+e))*tan(f*x+e)/c/f/(c^2-c^2*sec(f* x+e))^2
Time = 5.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.41 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=-\frac {a (1+\sec (e+f x)) \left (-58+33 \sec (e+f x)-12 \sec ^2(e+f x)+2 \sec ^3(e+f x)\right ) \tan (e+f x)}{315 c^5 f (-1+\sec (e+f x))^5} \]
-1/315*(a*(1 + Sec[e + f*x])*(-58 + 33*Sec[e + f*x] - 12*Sec[e + f*x]^2 + 2*Sec[e + f*x]^3)*Tan[e + f*x])/(c^5*f*(-1 + Sec[e + f*x])^5)
Time = 0.72 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)}{(c-c \sec (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5}dx\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)}{(c-c \sec (e+f x))^4}dx}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {\frac {2 \int \frac {\sec (e+f x) (\sec (e+f x) a+a)}{(c-c \sec (e+f x))^3}dx}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4}}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}dx}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4}}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)}{(c-c \sec (e+f x))^2}dx}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{5 f (c-c \sec (e+f x))^3}\right )}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4}}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2}dx}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{5 f (c-c \sec (e+f x))^3}\right )}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4}}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
| \(\Big \downarrow \) 4438 |
| \(\displaystyle \frac {\frac {2 \left (-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{15 c f (c-c \sec (e+f x))^2}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{5 f (c-c \sec (e+f x))^3}\right )}{7 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4}}{3 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}\) |
-1/9*((a + a*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^5) + (-1/ 7*((a + a*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^4) + (2*(-1/ 5*((a + a*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^3) - ((a + a *Sec[e + f*x])*Tan[e + f*x])/(15*c*f*(c - c*Sec[e + f*x])^2)))/(7*c))/(3*c )
3.1.9.3.1 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 = 0.90 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.39
| method | result | size |
| parallelrisch | \(\frac {a \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (35 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-135 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+189 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-105\right )}{2520 c^{5} f}\) | \(62\) |
| derivativedivides | \(\frac {a \left (\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}-\frac {3}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {3}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\right )}{8 f \,c^{5}}\) | \(63\) |
| default | \(\frac {a \left (\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}-\frac {3}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {3}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\right )}{8 f \,c^{5}}\) | \(63\) |
| risch | \(\frac {2 i a \left (315 \,{\mathrm e}^{8 i \left (f x +e \right )}-945 \,{\mathrm e}^{7 i \left (f x +e \right )}+2625 \,{\mathrm e}^{6 i \left (f x +e \right )}-3465 \,{\mathrm e}^{5 i \left (f x +e \right )}+3843 \,{\mathrm e}^{4 i \left (f x +e \right )}-2247 \,{\mathrm e}^{3 i \left (f x +e \right )}+1143 \,{\mathrm e}^{2 i \left (f x +e \right )}-207 \,{\mathrm e}^{i \left (f x +e \right )}+58\right )}{315 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9}}\) | \(114\) |
| norman | \(\frac {-\frac {a}{72 c f}+\frac {17 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{252 c f}-\frac {9 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{70 c f}+\frac {7 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{60 c f}-\frac {a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{24 c f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right ) c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\) | \(121\) |
1/2520*a*cot(1/2*f*x+1/2*e)^3*(35*cot(1/2*f*x+1/2*e)^6-135*cot(1/2*f*x+1/2 *e)^4+189*cot(1/2*f*x+1/2*e)^2-105)/c^5/f
Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=\frac {58 \, a \cos \left (f x + e\right )^{5} + 83 \, a \cos \left (f x + e\right )^{4} + 4 \, a \cos \left (f x + e\right )^{3} - 11 \, a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - 2 \, a}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )} \]
1/315*(58*a*cos(f*x + e)^5 + 83*a*cos(f*x + e)^4 + 4*a*cos(f*x + e)^3 - 11 *a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - 2*a)/((c^5*f*cos(f*x + e)^4 - 4*c^5 *f*cos(f*x + e)^3 + 6*c^5*f*cos(f*x + e)^2 - 4*c^5*f*cos(f*x + e) + c^5*f) *sin(f*x + e))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=- \frac {a \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{5}} \]
-a*(Integral(sec(e + f*x)/(sec(e + f*x)**5 - 5*sec(e + f*x)**4 + 10*sec(e + f*x)**3 - 10*sec(e + f*x)**2 + 5*sec(e + f*x) - 1), x) + Integral(sec(e + f*x)**2/(sec(e + f*x)**5 - 5*sec(e + f*x)**4 + 10*sec(e + f*x)**3 - 10*s ec(e + f*x)**2 + 5*sec(e + f*x) - 1), x))/c**5
Time = 0.22 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.25 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=-\frac {\frac {a {\left (\frac {180 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {378 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {5 \, a {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}}{5040 \, f} \]
-1/5040*(a*(180*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 378*sin(f*x + e)^4/( cos(f*x + e) + 1)^4 + 420*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 315*sin(f* x + e)^8/(cos(f*x + e) + 1)^8 - 35)*(cos(f*x + e) + 1)^9/(c^5*sin(f*x + e) ^9) + 5*a*(18*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 42*sin(f*x + e)^6/(cos (f*x + e) + 1)^6 + 63*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 7)*(cos(f*x + e) + 1)^9/(c^5*sin(f*x + e)^9))/f
Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.41 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=-\frac {105 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 189 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 135 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 35 \, a}{2520 \, c^{5} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} \]
-1/2520*(105*a*tan(1/2*f*x + 1/2*e)^6 - 189*a*tan(1/2*f*x + 1/2*e)^4 + 135 *a*tan(1/2*f*x + 1/2*e)^2 - 35*a)/(c^5*f*tan(1/2*f*x + 1/2*e)^9)
Time = 13.80 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx=\frac {a\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-135\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+189\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-105\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{2520\,c^5\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]