Integrand size = 34, antiderivative size = 117 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=-\frac {4 \sqrt {2} a^2 \arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f}+\frac {16 a^2 \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}-\frac {2 a^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 c f} \]
-4*a^2*arctan(1/2*c^(1/2)*tan(f*x+e)*2^(1/2)/(c-c*sec(f*x+e))^(1/2))*2^(1/ 2)/f/c^(1/2)+16/3*a^2*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-2/3*a^2*(c-c*sec (f*x+e))^(1/2)*tan(f*x+e)/c/f
Time = 0.56 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {2 a^2 \left (-6 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a (1+\sec (e+f x))}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {a (1+\sec (e+f x))} (7+\sec (e+f x))\right ) \tan (e+f x)}{3 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
(2*a^2*(-6*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a*(1 + Sec[e + f*x])]/(Sqrt[2]*Sqr t[a])] + Sqrt[a*(1 + Sec[e + f*x])]*(7 + Sec[e + f*x]))*Tan[e + f*x])/(3*f *Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.61 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {3042, 4444, 3042, 4444, 3042, 4282, 216}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^2}{\sqrt {c-c \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4444 |
| \(\displaystyle 2 a \int \frac {\sec (e+f x) (\sec (e+f x) a+a)}{\sqrt {c-c \sec (e+f x)}}dx+\frac {2 \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 4444 |
| \(\displaystyle 2 a \left (2 a \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}}dx+\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}\right )+\frac {2 \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}\right )+\frac {2 \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle 2 a \left (\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {4 a \int \frac {1}{\frac {c^2 \tan ^2(e+f x)}{c-c \sec (e+f x)}+2 c}d\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}}{f}\right )+\frac {2 \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f \sqrt {c-c \sec (e+f x)}}+2 a \left (\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {2 \sqrt {2} a \arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f}\right )\) |
(2*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]]) + 2*a*((-2*Sqrt[2]*a*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[ e + f*x]])])/(Sqrt[c]*f) + (2*a*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) )
3.1.75.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.))/ Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*d*Cot[e + f*x]*((c + d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[2*c*((2*n - 1)/(2*n - 1)) Int[Csc[e + f*x]*((c + d*Csc[e + f*x ])^(n - 1)/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Time = 4.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(-\frac {a^{2} \sqrt {2}\, \left (12 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \tan \left (f x +e \right )-7 \sqrt {2}\, \tan \left (f x +e \right )-\sqrt {2}\, \tan \left (f x +e \right ) \sec \left (f x +e \right )\right )}{3 f \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}}\) | \(108\) |
| parts | \(\frac {a^{2} \sqrt {2}\, \sin \left (f x +e \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}+\frac {a^{2} \sqrt {2}\, \left (-3 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \tan \left (f x +e \right )+\sqrt {2}\, \tan \left (f x +e \right )+\sqrt {2}\, \tan \left (f x +e \right ) \sec \left (f x +e \right )\right )}{3 f \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}}+\frac {2 a^{2} \sqrt {2}\, \left (-\sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )+\sqrt {2}\right ) \tan \left (f x +e \right )}{f \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}}\) | \(274\) |
-1/3*a^2/f*2^(1/2)/(-c*(sec(f*x+e)-1))^(1/2)*(12*arctan(1/2*2^(1/2)/(-cos( f*x+e)/(cos(f*x+e)+1))^(1/2))*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*tan(f*x+e )-7*2^(1/2)*tan(f*x+e)-2^(1/2)*tan(f*x+e)*sec(f*x+e))
Time = 0.32 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.93 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} a^{2} c \sqrt {-\frac {1}{c}} \cos \left (f x + e\right ) \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{c}} - {\left (3 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - {\left (7 \, a^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{3 \, c f \cos \left (f x + e\right ) \sin \left (f x + e\right )}, \frac {2 \, {\left (6 \, \sqrt {2} a^{2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (7 \, a^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{3 \, c f \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ] \]
[2/3*(3*sqrt(2)*a^2*c*sqrt(-1/c)*cos(f*x + e)*log(-(2*sqrt(2)*(cos(f*x + e )^2 + cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*sqrt(-1/c) - ( 3*cos(f*x + e) + 1)*sin(f*x + e))/((cos(f*x + e) - 1)*sin(f*x + e)))*sin(f *x + e) - (7*a^2*cos(f*x + e)^2 + 8*a^2*cos(f*x + e) + a^2)*sqrt((c*cos(f* x + e) - c)/cos(f*x + e)))/(c*f*cos(f*x + e)*sin(f*x + e)), 2/3*(6*sqrt(2) *a^2*sqrt(c)*arctan(sqrt(2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f* x + e)/(sqrt(c)*sin(f*x + e)))*cos(f*x + e)*sin(f*x + e) - (7*a^2*cos(f*x + e)^2 + 8*a^2*cos(f*x + e) + a^2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)) )/(c*f*cos(f*x + e)*sin(f*x + e))]
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{\sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{\sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \]
a**2*(Integral(sec(e + f*x)/sqrt(-c*sec(e + f*x) + c), x) + Integral(2*sec (e + f*x)**2/sqrt(-c*sec(e + f*x) + c), x) + Integral(sec(e + f*x)**3/sqrt (-c*sec(e + f*x) + c), x))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2} \sec \left (f x + e\right )}{\sqrt {-c \sec \left (f x + e\right ) + c}} \,d x } \]
Time = 1.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {4 \, a^{2} {\left (\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, c\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}}}\right )}}{3 \, f} \]
4/3*a^2*(3*sqrt(2)*arctan(sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)/sqrt(c))/sqrt (c) + sqrt(2)*(3*c*tan(1/2*f*x + 1/2*e)^2 - 4*c)/(c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2))/f
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{\cos \left (e+f\,x\right )\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}} \,d x \]