Integrand size = 34, antiderivative size = 128 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=-\frac {64 c^3 (a+a \sec (e+f x))^3 \tan (e+f x)}{693 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^2 (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{99 f}-\frac {2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{11 f} \]
-2/11*c*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f-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
Time = 1.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\frac {8 a^3 c^2 \cos ^6\left (\frac {1}{2} (e+f x)\right ) (277-364 \cos (e+f x)+151 \cos (2 (e+f x))) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x) \sqrt {c-c \sec (e+f x)}}{693 f} \]
(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 = 0.72 (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 |
| \(\displaystyle \frac {8}{11} c \left (-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3}{63 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}\) |
(-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
3.1.80.3.1 Defintions of rubi rules used
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 = 38.80 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.54
| method | result | size |
| default | \(\frac {2 c^{2} a^{3} \left (151 \cos \left (f x +e \right )^{2}-182 \cos \left (f x +e \right )+63\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right )^{4} \sec \left (f x +e \right )^{5} \csc \left (f x +e \right )}{693 f}\) | \(69\) |
| parts | \(\frac {2 a^{3} \left (\sec \left (f x +e \right )-1\right )^{2} \left (43 \cos \left (f x +e \right )^{2}-14 \cos \left (f x +e \right )+3\right ) c^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \csc \left (f x +e \right )}{15 f \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {2 a^{3} \left (1136 \cos \left (f x +e \right )^{5}-568 \cos \left (f x +e \right )^{4}+426 \cos \left (f x +e \right )^{3}-355 \cos \left (f x +e \right )^{2}+224 \cos \left (f x +e \right )-63\right ) \left (\sec \left (f x +e \right )-1\right )^{2} c^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \sec \left (f x +e \right )^{3} \csc \left (f x +e \right )}{693 f \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {2 a^{3} \left (46 \cos \left (f x +e \right )^{3}-23 \cos \left (f x +e \right )^{2}+12 \cos \left (f x +e \right )-3\right ) \left (\sec \left (f x +e \right )-1\right )^{2} c^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \sec \left (f x +e \right ) \csc \left (f x +e \right )}{7 f \left (\cos \left (f x +e \right )-1\right )^{2}}+\frac {2 a^{3} \left (584 \cos \left (f x +e \right )^{4}-292 \cos \left (f x +e \right )^{3}+219 \cos \left (f x +e \right )^{2}-130 \cos \left (f x +e \right )+35\right ) \left (\sec \left (f x +e \right )-1\right )^{2} c^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \sec \left (f x +e \right )^{2} \csc \left (f x +e \right )}{105 f \left (\cos \left (f x +e \right )-1\right )^{2}}\) | \(396\) |
2/693*c^2*a^3/f*(151*cos(f*x+e)^2-182*cos(f*x+e)+63)*(-c*(sec(f*x+e)-1))^( 1/2)*(cos(f*x+e)+1)^4*sec(f*x+e)^5*csc(f*x+e)
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.15 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\frac {2 \, {\left (151 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} + 422 \, a^{3} c^{2} \cos \left (f x + e\right )^{5} + 241 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} - 236 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} - 199 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{3} c^{2} \cos \left (f x + e\right ) + 63 \, a^{3} c^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{693 \, f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right )} \]
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))
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=a^{3} \left (\int c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )}\, dx + \int c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{5}{\left (e + f x \right )}\, dx + \int c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{6}{\left (e + f x \right )}\, dx\right ) \]
a**3*(Integral(c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x), x) + Integral( c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**2, x) + Integral(-2*c**2*sqrt (-c*sec(e + f*x) + c)*sec(e + f*x)**3, x) + Integral(-2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**4, x) + Integral(c**2*sqrt(-c*sec(e + f*x) + c) *sec(e + f*x)**5, x) + Integral(c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x )**6, x))
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\text {Timed out} \]
Time = 1.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx=\frac {64 \, \sqrt {2} {\left (99 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{4} + 154 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{5} + 63 \, c^{6}\right )} a^{3} c^{2}}{693 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {11}{2}} f} \]
64/693*sqrt(2)*(99*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^4 + 154*(c*tan(1/2*f *x + 1/2*e)^2 - c)*c^5 + 63*c^6)*a^3*c^2/((c*tan(1/2*f*x + 1/2*e)^2 - c)^( 11/2)*f)
Time = 25.00 (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=\frac {\left (\frac {a^3\,c^2\,2{}\mathrm {i}}{f}+\frac {a^3\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,302{}\mathrm {i}}{693\,f}\right )\,\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}}}}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}-\frac {\left (\frac {a^3\,c^2\,64{}\mathrm {i}}{11\,f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,64{}\mathrm {i}}{11\,f}\right )\,\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 ({\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 )}^5}+\frac {\left (\frac {a^3\,c^2\,16{}\mathrm {i}}{f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,944{}\mathrm {i}}{231\,f}\right )\,\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 ({\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 {\left (\frac {a^3\,c^2\,160{}\mathrm {i}}{9\,f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1120{}\mathrm {i}}{99\,f}\right )\,\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 ({\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 )}^4}-\frac {\left (\frac {a^3\,c^2\,20{}\mathrm {i}}{3\,f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,844{}\mathrm {i}}{693\,f}\right )\,\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 ({\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 )}-\frac {\left (\frac {a^3\,c^2\,160{}\mathrm {i}}{7\,f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,6880{}\mathrm {i}}{693\,f}\right )\,\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 ({\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} \]
(((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)