Integrand size = 34, antiderivative size = 128 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=-\frac {64 c^3 (a+a \sec (e+f x))^3 \tan (e+f x)}{693 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^2 (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{99 f}-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{11 f} \] Output:
-64/693*c^3*(a+a*sec(f*x+e))^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-16/99*c ^2*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/f-2/11*c*(a+a*sec( f*x+e))^3*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f
Time = 0.92 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\frac {8 a^3 c^2 \cos ^6\left (\frac {1}{2} (e+f x)\right ) (277-364 \cos (e+f x)+151 \cos (2 (e+f x))) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x) \sqrt {c-c \sec (e+f x)}}{693 f} \] Input:
Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2),x ]
Output:
(8*a^3*c^2*Cos[(e + f*x)/2]^6*(277 - 364*Cos[e + f*x] + 151*Cos[2*(e + f*x )])*Cot[(e + f*x)/2]*Sec[e + f*x]^5*Sqrt[c - c*Sec[e + f*x]])/(693*f)
Time = 0.73 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3042, 4443, 3042, 4443, 3042, 4441}
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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {8}{11} c \int \sec (e+f x) (\sec (e+f x) a+a)^3 (c-c \sec (e+f x))^{3/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8}{11} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {8}{11} c \left (\frac {4}{9} c \int \sec (e+f x) (\sec (e+f x) a+a)^3 \sqrt {c-c \sec (e+f x)}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8}{11} c \left (\frac {4}{9} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3 \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}\) |
| \(\Big \downarrow \) 4441 |
| \(\displaystyle \frac {8}{11} c \left (-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3}{63 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}\) |
Input:
Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2),x]
Output:
(-2*c*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(11* f) + (8*c*((-8*c^2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(63*f*Sqrt[c - c*S ec[e + f*x]]) - (2*c*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(9*f)))/11
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sq rt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[2*a*c*Cot[e + f *x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]])), x] / ; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[(-d)*Cot[e + f *x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(f*(m + n))), x] + Simp[c*((2*n - 1)/(m + n)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b *c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && !LtQ[m, -2^(-1)] && !(IGtQ[m - 1/2, 0] && LtQ[m, n])
Time = 53.79 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {64 a^{3} \sqrt {2}\, c^{2} \left (151 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-242 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+99\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{693 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5}}\) | \(113\) |
| parts | \(\frac {8 a^{3} \left (43 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-50 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {2}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}-\frac {8 a^{3} \left (9088 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-24992 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+28116 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-16401 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+5082 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-693\right ) \sqrt {2}\, \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{693 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5}}-\frac {8 a^{3} \left (92 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-161 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+98 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {2}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}+\frac {8 a^{3} \left (2336 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-5256 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+4599 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-1890 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+315\right ) \sqrt {2}\, \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{105 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}\) | \(484\) |
Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x,method=_RETURNV ERBOSE)
Output:
64/693*a^3*2^(1/2)*c^2/f*(151*cos(1/2*f*x+1/2*e)^4-242*cos(1/2*f*x+1/2*e)^ 2+99)*(-c/(2*cos(1/2*f*x+1/2*e)^2-1)*sin(1/2*f*x+1/2*e)^2)^(1/2)/(2*cos(1/ 2*f*x+1/2*e)^2-1)^5*cos(1/2*f*x+1/2*e)^6*cot(1/2*f*x+1/2*e)
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.15 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\frac {2 \, {\left (151 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} + 422 \, a^{3} c^{2} \cos \left (f x + e\right )^{5} + 241 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} - 236 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} - 199 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{3} c^{2} \cos \left (f x + e\right ) + 63 \, a^{3} c^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{693 \, f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right )} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x, algorith m="fricas")
Output:
2/693*(151*a^3*c^2*cos(f*x + e)^6 + 422*a^3*c^2*cos(f*x + e)^5 + 241*a^3*c ^2*cos(f*x + e)^4 - 236*a^3*c^2*cos(f*x + e)^3 - 199*a^3*c^2*cos(f*x + e)^ 2 + 70*a^3*c^2*cos(f*x + e) + 63*a^3*c^2)*sqrt((c*cos(f*x + e) - c)/cos(f* x + e))/(f*cos(f*x + e)^5*sin(f*x + e))
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**3*(c-c*sec(f*x+e))**(5/2),x)
Output:
Timed out
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x, algorith m="maxima")
Output:
Timed out
Time = 0.54 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\frac {64 \, \sqrt {2} {\left (99 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{6} + 154 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{7} + 63 \, c^{8}\right )} a^{3}}{693 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {11}{2}} f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x, algorith m="giac")
Output:
64/693*sqrt(2)*(99*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^6 + 154*(c*tan(1/2*f *x + 1/2*e)^2 - c)*c^7 + 63*c^8)*a^3/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(11/2 )*f)
Time = 22.76 (sec) , antiderivative size = 607, normalized size of antiderivative = 4.74 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
int(((a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^(5/2))/cos(e + f*x),x)
Output:
(((a^3*c^2*2i)/f + (a^3*c^2*exp(e*1i + f*x*1i)*302i)/(693*f))*(c - c/(exp( - e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/(exp(e*1i + f*x*1i) - 1 ) - (((a^3*c^2*64i)/(11*f) - (a^3*c^2*exp(e*1i + f*x*1i)*64i)/(11*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f* x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^5) + (((a^3*c^2*16i)/f - (a^3*c^2*exp( e*1i + f*x*1i)*944i)/(231*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^2) + (((a^3*c^2*160i)/(9*f) - (a^3*c^2*exp(e*1i + f*x*1i)*1120i)/(99*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x *1i) - 1)*(exp(e*2i + f*x*2i) + 1)^4) - (((a^3*c^2*20i)/(3*f) - (a^3*c^2*e xp(e*1i + f*x*1i)*844i)/(693*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)) - (((a^3*c^2*160i)/(7*f) - (a^3*c^2*exp(e*1i + f*x*1i)*6880i)/(693*f))*(c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/((exp(e*1i + f *x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^3)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\sqrt {c}\, a^{3} c^{2} \left (\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{6}d x +\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{5}d x -2 \left (\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{4}d x \right )-2 \left (\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{3}d x \right )+\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{2}d x +\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )d x \right ) \] Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x)
Output:
sqrt(c)*a**3*c**2*(int(sqrt( - sec(e + f*x) + 1)*sec(e + f*x)**6,x) + int( sqrt( - sec(e + f*x) + 1)*sec(e + f*x)**5,x) - 2*int(sqrt( - sec(e + f*x) + 1)*sec(e + f*x)**4,x) - 2*int(sqrt( - sec(e + f*x) + 1)*sec(e + f*x)**3, x) + int(sqrt( - sec(e + f*x) + 1)*sec(e + f*x)**2,x) + int(sqrt( - sec(e + f*x) + 1)*sec(e + f*x),x))