Integrand size = 34, antiderivative size = 142 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=\frac {128 c^4 \tan (e+f x)}{5 a f \sqrt {c-c \sec (e+f x)}}+\frac {32 c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{5 a f}+\frac {12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))} \]
12/5*c^2*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/a/f+2*c*(c-c*sec(f*x+e))^(5/2)* tan(f*x+e)/f/(a+a*sec(f*x+e))+128/5*c^4*tan(f*x+e)/a/f/(c-c*sec(f*x+e))^(1 /2)+32/5*c^3*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/a/f
Time = 0.85 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.51 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=\frac {2 c^4 \left (91+43 \sec (e+f x)-7 \sec ^2(e+f x)+\sec ^3(e+f x)\right ) \tan (e+f x)}{5 a f (1+\sec (e+f x)) \sqrt {c-c \sec (e+f x)}} \]
(2*c^4*(91 + 43*Sec[e + f*x] - 7*Sec[e + f*x]^2 + Sec[e + f*x]^3)*Tan[e + f*x])/(5*a*f*(1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])
Time = 0.69 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3042, 4442, 3042, 4280, 3042, 4280, 3042, 4279}
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.
\(\displaystyle \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a \sec (e+f x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}{a \csc \left (e+f x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 4442 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \int \sec (e+f x) (c-c \sec (e+f x))^{5/2}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}dx}{a}\) |
\(\Big \downarrow \) 4280 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \left (\frac {8}{5} c \int \sec (e+f x) (c-c \sec (e+f x))^{3/2}dx-\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \left (\frac {8}{5} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f}\right )}{a}\) |
\(\Big \downarrow \) 4280 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \left (\frac {8}{5} c \left (\frac {4}{3} c \int \sec (e+f x) \sqrt {c-c \sec (e+f x)}dx-\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 f}\right )-\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \left (\frac {8}{5} c \left (\frac {4}{3} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 f}\right )-\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f}\right )}{a}\) |
\(\Big \downarrow \) 4279 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}-\frac {6 c \left (\frac {8}{5} c \left (-\frac {8 c^2 \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 f}\right )-\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f}\right )}{a}\) |
(2*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) - ( 6*c*((-2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f) + (8*c*((-8*c^2* Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]]) - (2*c*Sqrt[c - c*Sec[e + f*x ]]*Tan[e + f*x])/(3*f)))/5))/a
3.1.86.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*b*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])), x] /; Free Q[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_ Symbol] :> Simp[(-b)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[a*((2*m - 1)/m) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && Intege rQ[2*m]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1))), x] - Simp[d*((2*n - 1)/(b*(2*m + 1))) Int[Csc[e + f*x]*(a + b*Csc[e + f* x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && LtQ[ m, -2^(-1)]
Time = 5.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.57
method | result | size |
default | \(\frac {2 c^{3} \left (\sec \left (f x +e \right )-1\right )^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (91 \cos \left (f x +e \right )^{3}+43 \cos \left (f x +e \right )^{2}-7 \cos \left (f x +e \right )+1\right ) \cot \left (f x +e \right )}{5 a f \left (\cos \left (f x +e \right )-1\right )^{3}}\) | \(81\) |
2/5/a/f*c^3*(sec(f*x+e)-1)^3*(-c*(sec(f*x+e)-1))^(1/2)*(91*cos(f*x+e)^3+43 *cos(f*x+e)^2-7*cos(f*x+e)+1)/(cos(f*x+e)-1)^3*cot(f*x+e)
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=-\frac {2 \, {\left (91 \, c^{3} \cos \left (f x + e\right )^{3} + 43 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{5 \, a f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \]
-2/5*(91*c^3*cos(f*x + e)^3 + 43*c^3*cos(f*x + e)^2 - 7*c^3*cos(f*x + e) + c^3)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/(a*f*cos(f*x + e)^2*sin(f*x + e))
Timed out. \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.15 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=\frac {8 \, {\left (16 \, \sqrt {2} c^{\frac {7}{2}} - \frac {56 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {70 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {35 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {5 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{5 \, a f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {7}{2}}} \]
8/5*(16*sqrt(2)*c^(7/2) - 56*sqrt(2)*c^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 70*sqrt(2)*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 35*sqrt( 2)*c^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 5*sqrt(2)*c^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)/(a*f*(sin(f*x + e)/(cos(f*x + e) + 1) + 1)^( 7/2)*(sin(f*x + e)/(cos(f*x + e) + 1) - 1)^(7/2))
Time = 0.83 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.76 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=-\frac {8 \, \sqrt {2} c^{3} {\left (\frac {5 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{a} - \frac {15 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c + 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{2} + c^{3}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} a}\right )}}{5 \, f} \]
-8/5*sqrt(2)*c^3*(5*sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)/a - (15*(c*tan(1/2* f*x + 1/2*e)^2 - c)^2*c + 5*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^2 + c^3)/((c* tan(1/2*f*x + 1/2*e)^2 - c)^(5/2)*a))/f
Time = 17.98 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.15 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx=-\frac {2\,c^3\,\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,86{}\mathrm {i}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,245{}\mathrm {i}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,180{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,245{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,86{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,91{}\mathrm {i}+91{}\mathrm {i}\right )}{5\,a\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2} \]