Integrand size = 32, antiderivative size = 122 \[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=-\frac {64 c^3 (a+a \sec (e+f x)) \tan (e+f x)}{105 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^2 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{35 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{7 f} \] Output:
-64/105*c^3*(a+a*sec(f*x+e))*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-16/35*c^2 *(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/f-2/7*c*(a+a*sec(f*x+e ))*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f
Time = 0.59 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.49 \[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=-\frac {2 a c^3 (1+\sec (e+f x)) \left (71-54 \sec (e+f x)+15 \sec ^2(e+f x)\right ) \tan (e+f x)}{105 f \sqrt {c-c \sec (e+f x)}} \] Input:
Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2),x]
Output:
(-2*a*c^3*(1 + Sec[e + f*x])*(71 - 54*Sec[e + f*x] + 15*Sec[e + f*x]^2)*Ta n[e + f*x])/(105*f*Sqrt[c - c*Sec[e + f*x]])
Time = 0.60 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, 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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {8}{7} c \int \sec (e+f x) (\sec (e+f x) a+a) (c-c \sec (e+f x))^{3/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8}{7} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f}\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {8}{7} c \left (\frac {4}{5} c \int \sec (e+f x) (\sec (e+f x) a+a) \sqrt {c-c \sec (e+f x)}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{5 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8}{7} c \left (\frac {4}{5} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right ) \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{5 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f}\) |
| \(\Big \downarrow \) 4441 |
| \(\displaystyle \frac {8}{7} c \left (-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)}{15 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{5 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f}\) |
Input:
Int[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2),x]
Output:
(-2*c*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(7*f) + (8*c*((-8*c^2*(a + a*Sec[e + f*x])*Tan[e + f*x])/(15*f*Sqrt[c - c*Sec[e + f*x]]) - (2*c*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x] )/(5*f)))/7
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 = 4.95 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.75
| method | result | size |
| default | \(a \left (-\frac {8 \left (92 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-161 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+98 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {2}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}+\frac {8 \left (43 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-50 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {2}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}\right )\) | \(213\) |
| parts | \(\frac {8 a \left (43 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-50 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {2}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}-\frac {8 a \left (92 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-161 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+98 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21\right ) \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {2}\, c^{2} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}\) | \(213\) |
Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2),x,method=_RETURNVER BOSE)
Output:
a*(-8/21/f*(92*cos(1/2*f*x+1/2*e)^6-161*cos(1/2*f*x+1/2*e)^4+98*cos(1/2*f* x+1/2*e)^2-21)*(-c/(2*cos(1/2*f*x+1/2*e)^2-1)*sin(1/2*f*x+1/2*e)^2)^(1/2)* 2^(1/2)*c^2/(2*cos(1/2*f*x+1/2*e)^2-1)^3*cot(1/2*f*x+1/2*e)+8/15/f*(43*cos (1/2*f*x+1/2*e)^4-50*cos(1/2*f*x+1/2*e)^2+15)*(-c/(2*cos(1/2*f*x+1/2*e)^2- 1)*sin(1/2*f*x+1/2*e)^2)^(1/2)*2^(1/2)*c^2/(2*cos(1/2*f*x+1/2*e)^2-1)^2*co t(1/2*f*x+1/2*e))
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.86 \[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=\frac {2 \, {\left (71 \, a c^{2} \cos \left (f x + e\right )^{4} + 88 \, a c^{2} \cos \left (f x + e\right )^{3} - 22 \, a c^{2} \cos \left (f x + e\right )^{2} - 24 \, a c^{2} \cos \left (f x + e\right ) + 15 \, a c^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{105 \, f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right )} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2),x, algorithm= "fricas")
Output:
2/105*(71*a*c^2*cos(f*x + e)^4 + 88*a*c^2*cos(f*x + e)^3 - 22*a*c^2*cos(f* x + e)^2 - 24*a*c^2*cos(f*x + e) + 15*a*c^2)*sqrt((c*cos(f*x + e) - c)/cos (f*x + e))/(f*cos(f*x + e)^3*sin(f*x + e))
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \sec \left (f x + e\right ) \,d x } \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2),x, algorithm= "maxima")
Output:
-2/105*(105*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(3/4)*(3*(a*c^2*f*cos(2*f*x + 2*e)^2 + a*c^2*f*sin(2*f*x + 2*e)^2 + 2 *a*c^2*f*cos(2*f*x + 2*e) + a*c^2*f)*integrate((((cos(6*f*x + 6*e)*cos(2*f *x + 2*e) + 2*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin (6*f*x + 6*e)*sin(2*f*x + 2*e) + 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin (2*f*x + 2*e)^2)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (c os(2*f*x + 2*e)*sin(6*f*x + 6*e) + 2*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - c os(6*f*x + 6*e)*sin(2*f*x + 2*e) - 2*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*si n(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(5/2*arctan2(sin(2* f*x + 2*e), cos(2*f*x + 2*e) + 1)) - ((cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 2*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - cos(6*f*x + 6*e)*sin(2*f*x + 2*e) - 2*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*cos(7/2*arctan2(sin(2*f*x + 2*e), co s(2*f*x + 2*e))) - (cos(6*f*x + 6*e)*cos(2*f*x + 2*e) + 2*cos(4*f*x + 4*e) *cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin(2*f*x + 2*e)^2)*sin(7/2*arcta n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)))/(((cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + (cos (2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*cos(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e)^2 + 2*cos(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^2 + ...
Time = 0.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66 \[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=\frac {16 \, \sqrt {2} {\left (35 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{4} + 42 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{5} + 15 \, c^{6}\right )} a}{105 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {7}{2}} f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2),x, algorithm= "giac")
Output:
16/105*sqrt(2)*(35*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^4 + 42*(c*tan(1/2*f* x + 1/2*e)^2 - c)*c^5 + 15*c^6)*a/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(7/2)*f)
Time = 14.39 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.15 \[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,2{}\mathrm {i}}{f}+\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,142{}\mathrm {i}}{105\,f}\right )}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,16{}\mathrm {i}}{7\,f}-\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,16{}\mathrm {i}}{7\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,8{}\mathrm {i}}{5\,f}-\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,184{}\mathrm {i}}{35\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,4{}\mathrm {i}}{3\,f}+\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,244{}\mathrm {i}}{105\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \] Input:
int(((a + a/cos(e + f*x))*(c - c/cos(e + f*x))^(5/2))/cos(e + f*x),x)
Output:
((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^2*2i) /f + (a*c^2*exp(e*1i + f*x*1i)*142i)/(105*f)))/(exp(e*1i + f*x*1i) - 1) + ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^2*16i )/(7*f) - (a*c^2*exp(e*1i + f*x*1i)*16i)/(7*f)))/((exp(e*1i + f*x*1i) - 1) *(exp(e*2i + f*x*2i) + 1)^3) - ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^2*8i)/(5*f) - (a*c^2*exp(e*1i + f*x*1i)*184i)/(3 5*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^2) - ((c - c/(ex p(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^2*4i)/(3*f) + (a *c^2*exp(e*1i + f*x*1i)*244i)/(105*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2 i + f*x*2i) + 1))
\[ \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx=\sqrt {c}\, a \,c^{2} \left (\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{4}d x -\left (\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{3}d x \right )-\left (\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{2}d x \right )+\int \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )d x \right ) \] Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2),x)
Output:
sqrt(c)*a*c**2*(int(sqrt( - sec(e + f*x) + 1)*sec(e + f*x)**4,x) - int(sqr t( - 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))