Integrand size = 34, antiderivative size = 123 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=-\frac {16 c^3 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2} \]
2/3*c*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f/(a+a*sec(f*x+e))^2-16/3*c^3*tan( f*x+e)/a^2/f/(c-c*sec(f*x+e))^(1/2)-8/3*c^2*(c-c*sec(f*x+e))^(1/2)*tan(f*x +e)/f/(a^2+a^2*sec(f*x+e))
Time = 0.90 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=-\frac {2 c^3 \left (11+18 \sec (e+f x)+3 \sec ^2(e+f x)\right ) \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}} \]
(-2*c^3*(11 + 18*Sec[e + f*x] + 3*Sec[e + f*x]^2)*Tan[e + f*x])/(3*a^2*f*( 1 + Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])
Time = 0.70 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3042, 4442, 3042, 4442, 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.
\(\displaystyle \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a \sec (e+f x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
\(\Big \downarrow \) 4442 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2}-\frac {4 c \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{\sec (e+f x) a+a}dx}{3 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2}-\frac {4 c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx}{3 a}\) |
\(\Big \downarrow \) 4442 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2}-\frac {4 c \left (\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f (a \sec (e+f x)+a)}-\frac {2 c \int \sec (e+f x) \sqrt {c-c \sec (e+f x)}dx}{a}\right )}{3 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2}-\frac {4 c \left (\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f (a \sec (e+f x)+a)}-\frac {2 c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a}\right )}{3 a}\) |
\(\Big \downarrow \) 4279 |
\(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2}-\frac {4 c \left (\frac {4 c^2 \tan (e+f x)}{a f \sqrt {c-c \sec (e+f x)}}+\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f (a \sec (e+f x)+a)}\right )}{3 a}\) |
(2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (4*c*((4*c^2*Tan[e + f*x])/(a*f*Sqrt[c - c*Sec[e + f*x]]) + (2*c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))))/(3*a)
3.1.94.3.1 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_)*(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 = 12.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64
method | result | size |
default | \(-\frac {2 \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right )^{2} c^{2} \left (11 \cos \left (f x +e \right )^{2}+18 \cos \left (f x +e \right )+3\right ) \cot \left (f x +e \right )^{2} \csc \left (f x +e \right )}{3 a^{2} f \left (\cos \left (f x +e \right )-1\right )}\) | \(79\) |
-2/3/a^2/f*(-c*(sec(f*x+e)-1))^(1/2)*(sec(f*x+e)-1)^2*c^2*(11*cos(f*x+e)^2 +18*cos(f*x+e)+3)/(cos(f*x+e)-1)*cot(f*x+e)^2*csc(f*x+e)
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=\frac {2 \, {\left (11 \, c^{2} \cos \left (f x + e\right )^{2} + 18 \, c^{2} \cos \left (f x + e\right ) + 3 \, c^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \sin \left (f x + e\right )} \]
2/3*(11*c^2*cos(f*x + e)^2 + 18*c^2*cos(f*x + e) + 3*c^2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/((a^2*f*cos(f*x + e) + a^2*f)*sin(f*x + e))
Timed out. \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.33 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=\frac {2 \, {\left (8 \, \sqrt {2} c^{\frac {5}{2}} - \frac {20 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {2 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {\sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{3 \, a^{2} f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {5}{2}}} \]
2/3*(8*sqrt(2)*c^(5/2) - 20*sqrt(2)*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*sqrt(2)*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 2*sqrt(2) *c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - sqrt(2)*c^(5/2)*sin(f*x + e )^8/(cos(f*x + e) + 1)^8)/(a^2*f*(sin(f*x + e)/(cos(f*x + e) + 1) + 1)^(5/ 2)*(sin(f*x + e)/(cos(f*x + e) + 1) - 1)^(5/2))
Time = 0.81 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=-\frac {2 \, \sqrt {2} c^{2} {\left (\frac {3 \, c}{\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a^{2}} - \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} a^{4} c^{2} + 6 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a^{4} c^{3}}{a^{6} c^{3}}\right )}}{3 \, f} \]
-2/3*sqrt(2)*c^2*(3*c/(sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)*a^2) - ((c*tan(1 /2*f*x + 1/2*e)^2 - c)^(3/2)*a^4*c^2 + 6*sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c )*a^4*c^3)/(a^6*c^3))/f
Time = 17.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx=\frac {2\,c^2\,\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,36{}\mathrm {i}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,34{}\mathrm {i}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,36{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,11{}\mathrm {i}+11{}\mathrm {i}\right )}{3\,a^2\,f\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}^3} \]