Integrand size = 36, antiderivative size = 42 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}} \]
Time = 0.86 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {(a (1+\sec (e+f x)))^{3/2} \tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}} \]
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3042, 4438}
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.
\(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4438 |
\(\displaystyle -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}}\) |
3.2.19.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] /; Fre eQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] & & EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
Time = 3.59 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.55
method | result | size |
default | \(-\frac {a \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\tan \left (f x +e \right )+\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )}{4 f \left (\sec \left (f x +e \right )-1\right )^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, c^{2}}\) | \(65\) |
risch | \(\frac {2 i a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{3 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{3} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(116\) |
-1/4/f*a*(a*(sec(f*x+e)+1))^(1/2)/(sec(f*x+e)-1)^2/(-c*(sec(f*x+e)-1))^(1/ 2)/c^2*(tan(f*x+e)+sec(f*x+e)*tan(f*x+e))
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.26 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\frac {a \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2}}{{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )} \]
a*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f* x + e))*cos(f*x + e)^2/((c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) + c^3 *f)*sin(f*x + e))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sec {\left (e + f x \right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (36) = 72\).
Time = 0.42 (sec) , antiderivative size = 533, normalized size of antiderivative = 12.69 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\frac {2 \, {\left (6 \, a \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 6 \, a \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, a \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - {\left (a \sin \left (3 \, f x + 3 \, e\right ) + a \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (a \cos \left (3 \, f x + 3 \, e\right ) + a \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - {\left (6 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} \sin \left (3 \, f x + 3 \, e\right ) - a \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (c^{3} \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, c^{3} \cos \left (f x + e\right )^{2} + c^{3} \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \sin \left (2 \, f x + 2 \, e\right )^{2} - 48 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, c^{3} \sin \left (f x + e\right )^{2} - 8 \, c^{3} \cos \left (f x + e\right ) + c^{3} - 2 \, {\left (4 \, c^{3} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 8 \, {\left (6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 12 \, {\left (4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (2 \, c^{3} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 16 \, {\left (3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) - 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \]
2*(6*a*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) + 6*a*cos(f*x + e)*sin(2*f*x + 2* e) - 6*a*cos(2*f*x + 2*e)*sin(f*x + e) - (a*sin(3*f*x + 3*e) + a*sin(f*x + e))*cos(4*f*x + 4*e) + (a*cos(3*f*x + 3*e) + a*cos(f*x + e))*sin(4*f*x + 4*e) - (6*a*cos(2*f*x + 2*e) + a)*sin(3*f*x + 3*e) - a*sin(f*x + e))*sqrt( a)*sqrt(c)/((c^3*cos(4*f*x + 4*e)^2 + 16*c^3*cos(3*f*x + 3*e)^2 + 36*c^3*c os(2*f*x + 2*e)^2 + 16*c^3*cos(f*x + e)^2 + c^3*sin(4*f*x + 4*e)^2 + 16*c^ 3*sin(3*f*x + 3*e)^2 + 36*c^3*sin(2*f*x + 2*e)^2 - 48*c^3*sin(2*f*x + 2*e) *sin(f*x + e) + 16*c^3*sin(f*x + e)^2 - 8*c^3*cos(f*x + e) + c^3 - 2*(4*c^ 3*cos(3*f*x + 3*e) - 6*c^3*cos(2*f*x + 2*e) + 4*c^3*cos(f*x + e) - c^3)*co s(4*f*x + 4*e) - 8*(6*c^3*cos(2*f*x + 2*e) - 4*c^3*cos(f*x + e) + c^3)*cos (3*f*x + 3*e) - 12*(4*c^3*cos(f*x + e) - c^3)*cos(2*f*x + 2*e) - 4*(2*c^3* sin(3*f*x + 3*e) - 3*c^3*sin(2*f*x + 2*e) + 2*c^3*sin(f*x + e))*sin(4*f*x + 4*e) - 16*(3*c^3*sin(2*f*x + 2*e) - 2*c^3*sin(f*x + e))*sin(3*f*x + 3*e) )*f)
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Time = 16.76 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.93 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {2\,a\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (6\,\sin \left (e+f\,x\right )-8\,\sin \left (2\,e+2\,f\,x\right )+7\,\sin \left (3\,e+3\,f\,x\right )-4\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (5\,e+5\,f\,x\right )\right )}{c^3\,f\,\left (48\,\cos \left (e+f\,x\right )+15\,\cos \left (2\,e+2\,f\,x\right )-40\,\cos \left (3\,e+3\,f\,x\right )+26\,\cos \left (4\,e+4\,f\,x\right )-8\,\cos \left (5\,e+5\,f\,x\right )+\cos \left (6\,e+6\,f\,x\right )-42\right )} \]
-(2*a*((a*(cos(e + f*x) + 1))/cos(e + f*x))^(1/2)*((c*(cos(e + f*x) - 1))/ cos(e + f*x))^(1/2)*(6*sin(e + f*x) - 8*sin(2*e + 2*f*x) + 7*sin(3*e + 3*f *x) - 4*sin(4*e + 4*f*x) + sin(5*e + 5*f*x)))/(c^3*f*(48*cos(e + f*x) + 15 *cos(2*e + 2*f*x) - 40*cos(3*e + 3*f*x) + 26*cos(4*e + 4*f*x) - 8*cos(5*e + 5*f*x) + cos(6*e + 6*f*x) - 42))