Integrand size = 34, antiderivative size = 171 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=-\frac {256 c^4 (a+a \sec (e+f x))^2 \tan (e+f x)}{1155 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 (a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{231 f}-\frac {8 c^2 (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{33 f}-\frac {2 c (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{11 f} \]
-8/33*c^2*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f-2/11*c*(a +a*sec(f*x+e))^2*(c-c*sec(f*x+e))^(5/2)*tan(f*x+e)/f-256/1155*c^4*(a+a*sec (f*x+e))^2*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-64/231*c^3*(a+a*sec(f*x+e)) ^2*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/f
Time = 2.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=\frac {2 a^2 c^3 \cos ^4\left (\frac {1}{2} (e+f x)\right ) (-1930+3419 \cos (e+f x)-1510 \cos (2 (e+f x))+533 \cos (3 (e+f x))) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x) \sqrt {c-c \sec (e+f x)}}{1155 f} \]
(2*a^2*c^3*Cos[(e + f*x)/2]^4*(-1930 + 3419*Cos[e + f*x] - 1510*Cos[2*(e + f*x)] + 533*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^5*Sqrt[c - c* Sec[e + f*x]])/(1155*f)
Time = 0.95 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3042, 4443, 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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}dx\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {12}{11} c \int \sec (e+f x) (\sec (e+f x) a+a)^2 (c-c \sec (e+f x))^{5/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {12}{11} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {12}{11} c \left (\frac {8}{9} c \int \sec (e+f x) (\sec (e+f x) a+a)^2 (c-c \sec (e+f x))^{3/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {12}{11} c \left (\frac {8}{9} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
| \(\Big \downarrow \) 4443 |
| \(\displaystyle \frac {12}{11} c \left (\frac {8}{9} c \left (\frac {4}{7} c \int \sec (e+f x) (\sec (e+f x) a+a)^2 \sqrt {c-c \sec (e+f x)}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt {c-c \sec (e+f x)}}{7 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {12}{11} c \left (\frac {8}{9} c \left (\frac {4}{7} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2 \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt {c-c \sec (e+f x)}}{7 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
| \(\Big \downarrow \) 4441 |
| \(\displaystyle \frac {12}{11} c \left (\frac {8}{9} c \left (-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^2}{35 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt {c-c \sec (e+f x)}}{7 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f}\right )-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}\) |
(-2*c*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(11* f) + (12*c*((-2*c*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(9*f) + (8*c*((-8*c^2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(35*f*S qrt[c - c*Sec[e + f*x]]) - (2*c*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(7*f)))/9))/11
3.1.71.3.1 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 = 16.90 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.46
| method | result | size |
| default | \(\frac {2 a^{2} c^{3} \left (533 \cos \left (f x +e \right )^{3}-755 \cos \left (f x +e \right )^{2}+455 \cos \left (f x +e \right )-105\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right )^{3} \sec \left (f x +e \right )^{5} \csc \left (f x +e \right )}{1155 f}\) | \(79\) |
| parts | \(-\frac {2 a^{2} \left (\sec \left (f x +e \right )-1\right )^{3} \left (177 \cos \left (f x +e \right )^{3}-71 \cos \left (f x +e \right )^{2}+27 \cos \left (f x +e \right )-5\right ) c^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \csc \left (f x +e \right )}{35 f \left (\cos \left (f x +e \right )-1\right )^{3}}-\frac {2 a^{2} \left (12104 \cos \left (f x +e \right )^{5}-6052 \cos \left (f x +e \right )^{4}+4539 \cos \left (f x +e \right )^{3}-3205 \cos \left (f x +e \right )^{2}+1505 \cos \left (f x +e \right )-315\right ) \left (\sec \left (f x +e \right )-1\right )^{3} c^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \sec \left (f x +e \right )^{2} \csc \left (f x +e \right )}{3465 f \left (\cos \left (f x +e \right )-1\right )^{3}}+\frac {4 a^{2} \left (182 \cos \left (f x +e \right )^{4}-91 \cos \left (f x +e \right )^{3}+57 \cos \left (f x +e \right )^{2}-25 \cos \left (f x +e \right )+5\right ) \left (\sec \left (f x +e \right )-1\right )^{3} c^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \sec \left (f x +e \right ) \csc \left (f x +e \right )}{45 f \left (\cos \left (f x +e \right )-1\right )^{3}}\) | \(310\) |
2/1155*a^2*c^3/f*(533*cos(f*x+e)^3-755*cos(f*x+e)^2+455*cos(f*x+e)-105)*(- c*(sec(f*x+e)-1))^(1/2)*(cos(f*x+e)+1)^3*sec(f*x+e)^5*csc(f*x+e)
Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.86 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=\frac {2 \, {\left (533 \, a^{2} c^{3} \cos \left (f x + e\right )^{6} + 844 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} - 211 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} - 472 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} + 295 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} + 140 \, a^{2} c^{3} \cos \left (f x + e\right ) - 105 \, a^{2} c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{1155 \, f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right )} \]
2/1155*(533*a^2*c^3*cos(f*x + e)^6 + 844*a^2*c^3*cos(f*x + e)^5 - 211*a^2* c^3*cos(f*x + e)^4 - 472*a^2*c^3*cos(f*x + e)^3 + 295*a^2*c^3*cos(f*x + e) ^2 + 140*a^2*c^3*cos(f*x + e) - 105*a^2*c^3)*sqrt((c*cos(f*x + e) - c)/cos (f*x + e))/(f*cos(f*x + e)^5*sin(f*x + e))
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=\text {Timed out} \]
Time = 1.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.64 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=-\frac {64 \, \sqrt {2} {\left (231 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} c^{3} + 495 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{4} + 385 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{5} + 105 \, c^{6}\right )} a^{2} c^{3}}{1155 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {11}{2}} f} \]
-64/1155*sqrt(2)*(231*(c*tan(1/2*f*x + 1/2*e)^2 - c)^3*c^3 + 495*(c*tan(1/ 2*f*x + 1/2*e)^2 - c)^2*c^4 + 385*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^5 + 105 *c^6)*a^2*c^3/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(11/2)*f)
Time = 25.26 (sec) , antiderivative size = 606, normalized size of antiderivative = 3.54 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx=\frac {\left (\frac {a^2\,c^3\,2{}\mathrm {i}}{f}+\frac {a^2\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1066{}\mathrm {i}}{1155\,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^2\,c^3\,64{}\mathrm {i}}{11\,f}-\frac {a^2\,c^3\,{\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^2\,c^3\,32{}\mathrm {i}}{3\,f}-\frac {a^2\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,608{}\mathrm {i}}{33\,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^2\,c^3\,4{}\mathrm {i}}{f}+\frac {a^2\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,2932{}\mathrm {i}}{1155\,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^2\,c^3\,16{}\mathrm {i}}{5\,f}+\frac {a^2\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,4272{}\mathrm {i}}{385\,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^2\,c^3\,32{}\mathrm {i}}{7\,f}-\frac {a^2\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,4640{}\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 )}^3} \]
(((a^2*c^3*2i)/f + (a^2*c^3*exp(e*1i + f*x*1i)*1066i)/(1155*f))*(c - c/(ex p(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2))/(exp(e*1i + f*x*1i) - 1) + (((a^2*c^3*64i)/(11*f) - (a^2*c^3*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^2*c^3*32i)/(3*f) - (a^2*c^ 3*exp(e*1i + f*x*1i)*608i)/(33*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^2*c^3*4i)/f + (a^2*c^3*exp(e*1i + f*x*1i)*2932i)/(1155*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^2*c^3*16i)/(5*f) + (a^2*c^3*ex p(e*1i + f*x*1i)*4272i)/(385*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^2*c^3*32i)/(7*f) - (a^2*c^3*exp(e*1i + f*x*1i)*4640i)/(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)^3)