Integrand size = 36, antiderivative size = 89 \[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=-\frac {c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}-\frac {c (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 f} \]
-1/3*c^2*(a+a*sec(f*x+e))^(3/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-1/3*c* (a+a*sec(f*x+e))^(3/2)*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/f
Time = 0.70 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \sec (e+f x) \left (-3+\sec ^2(e+f x)\right ) \tan (e+f x)}{3 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
(a^2*c^2*Sec[e + f*x]*(-3 + Sec[e + f*x]^2)*Tan[e + f*x])/(3*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.57 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
\(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^{3/2} (c-c \sec (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
\(\Big \downarrow \) 4443 |
\(\displaystyle \frac {2}{3} c \int \sec (e+f x) (\sec (e+f x) a+a)^{3/2} \sqrt {c-c \sec (e+f x)}dx-\frac {c \tan (e+f x) (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {c \tan (e+f x) (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}}{3 f}\) |
\(\Big \downarrow \) 4441 |
\(\displaystyle -\frac {c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{3 f \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x) (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}}{3 f}\) |
-1/3*(c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f *x]]) - (c*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x ])/(3*f)
3.2.15.3.1 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 = 3.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {a \left (\sec \left (f x +e \right )-1\right ) \left (2 \cos \left (f x +e \right )-1\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, c \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\cos \left (f x +e \right )+1\right )^{2} \sec \left (f x +e \right ) \csc \left (f x +e \right )}{3 f \left (\cos \left (f x +e \right )-1\right )}\) | \(83\) |
risch | \(\frac {2 i a c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}+2 \,{\mathrm e}^{3 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\) | \(142\) |
-1/3/f*a*(sec(f*x+e)-1)*(2*cos(f*x+e)-1)*(-c*(sec(f*x+e)-1))^(1/2)*c*(a*(s ec(f*x+e)+1))^(1/2)*(cos(f*x+e)+1)^2/(cos(f*x+e)-1)*sec(f*x+e)*csc(f*x+e)
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {{\left (3 \, a c \cos \left (f x + e\right )^{2} - a c\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \]
1/3*(3*a*c*cos(f*x + e)^2 - a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*s qrt((c*cos(f*x + e) - c)/cos(f*x + e))/(f*cos(f*x + e)^2*sin(f*x + e))
Timed out. \[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (77) = 154\).
Time = 0.42 (sec) , antiderivative size = 550, normalized size of antiderivative = 6.18 \[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {2 \, {\left (6 \, a c \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a c \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 9 \, a c \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - 3 \, a c \sin \left (f x + e\right ) - {\left (3 \, a c \sin \left (5 \, f x + 5 \, e\right ) + 2 \, a c \sin \left (3 \, f x + 3 \, e\right ) + 3 \, a c \sin \left (f x + e\right )\right )} \cos \left (6 \, f x + 6 \, e\right ) + 9 \, {\left (a c \sin \left (4 \, f x + 4 \, e\right ) + a c \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (5 \, f x + 5 \, e\right ) - 3 \, {\left (2 \, a c \sin \left (3 \, f x + 3 \, e\right ) + 3 \, a c \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (3 \, a c \cos \left (5 \, f x + 5 \, e\right ) + 2 \, a c \cos \left (3 \, f x + 3 \, e\right ) + 3 \, a c \cos \left (f x + e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (3 \, a c \cos \left (4 \, f x + 4 \, e\right ) + 3 \, a c \cos \left (2 \, f x + 2 \, e\right ) + a c\right )} \sin \left (5 \, f x + 5 \, e\right ) + 3 \, {\left (2 \, a c \cos \left (3 \, f x + 3 \, e\right ) + 3 \, a c \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (3 \, a c \cos \left (2 \, f x + 2 \, e\right ) + a c\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} \sqrt {a} \sqrt {c}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, f x + 4 \, e\right ) + 3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (6 \, f x + 6 \, e\right ) + \cos \left (6 \, f x + 6 \, e\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + 9 \, \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 6 \, {\left (\sin \left (4 \, f x + 4 \, e\right ) + \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) + \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, \sin \left (4 \, f x + 4 \, e\right )^{2} + 18 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \]
2/3*(6*a*c*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) + 9*a*c*cos(f*x + e)*sin(2*f* x + 2*e) - 9*a*c*cos(2*f*x + 2*e)*sin(f*x + e) - 3*a*c*sin(f*x + e) - (3*a *c*sin(5*f*x + 5*e) + 2*a*c*sin(3*f*x + 3*e) + 3*a*c*sin(f*x + e))*cos(6*f *x + 6*e) + 9*(a*c*sin(4*f*x + 4*e) + a*c*sin(2*f*x + 2*e))*cos(5*f*x + 5* e) - 3*(2*a*c*sin(3*f*x + 3*e) + 3*a*c*sin(f*x + e))*cos(4*f*x + 4*e) + (3 *a*c*cos(5*f*x + 5*e) + 2*a*c*cos(3*f*x + 3*e) + 3*a*c*cos(f*x + e))*sin(6 *f*x + 6*e) - 3*(3*a*c*cos(4*f*x + 4*e) + 3*a*c*cos(2*f*x + 2*e) + a*c)*si n(5*f*x + 5*e) + 3*(2*a*c*cos(3*f*x + 3*e) + 3*a*c*cos(f*x + e))*sin(4*f*x + 4*e) - 2*(3*a*c*cos(2*f*x + 2*e) + a*c)*sin(3*f*x + 3*e))*sqrt(a)*sqrt( c)/((2*(3*cos(4*f*x + 4*e) + 3*cos(2*f*x + 2*e) + 1)*cos(6*f*x + 6*e) + co s(6*f*x + 6*e)^2 + 6*(3*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 9*cos(4*f *x + 4*e)^2 + 9*cos(2*f*x + 2*e)^2 + 6*(sin(4*f*x + 4*e) + sin(2*f*x + 2*e ))*sin(6*f*x + 6*e) + sin(6*f*x + 6*e)^2 + 9*sin(4*f*x + 4*e)^2 + 18*sin(4 *f*x + 4*e)*sin(2*f*x + 2*e) + 9*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e) + 1)*f)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \sec \left (f x + e\right ) \,d x } \]
Time = 14.72 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.21 \[ \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {2\,a\,c\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\left (2\,\sin \left (e+f\,x\right )+5\,\sin \left (3\,e+3\,f\,x\right )+3\,\sin \left (5\,e+5\,f\,x\right )\right )}{3\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-2\,\cos \left (4\,e+4\,f\,x\right )-\cos \left (6\,e+6\,f\,x\right )+2\right )} \]