Integrand size = 15, antiderivative size = 87 \[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d} \]
2*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/d/a^(1/2)-arctan(1/2*a ^(1/2)*tan(d*x+c)*2^(1/2)/(a-a*sec(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=-\frac {i \left (-1+e^{i (c+d x)}\right ) \left (\text {arcsinh}\left (e^{i (c+d x)}\right )-\sqrt {2} \text {arctanh}\left (\frac {1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right )}{d \sqrt {1+e^{2 i (c+d x)}} \sqrt {a-a \sec (c+d x)}} \]
((-I)*(-1 + E^(I*(c + d*x)))*(ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*ArcTanh[( 1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + ArcTanh[Sq rt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a *Sec[c + d*x]])
Time = 0.39 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 4263, 3042, 4261, 216, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4263 |
| \(\displaystyle \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}}dx+\frac {\int \sqrt {a-a \sec (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\int \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {2 \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}+\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {2 \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+2 a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}\) |
(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])] )/(Sqrt[a]*d)
3.2.42.3.1 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[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[1/a I nt[Sqrt[a + b*Csc[c + d*x]], x], x] - Simp[b/a Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 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]
Time = 0.89 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sin \left (d x +c \right ) \left (\arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+\sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\) | \(109\) |
1/d*2^(1/2)*sin(d*x+c)*(arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1 /2))+2^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)))/(cos(d*x+c)+1)/(- a*(sec(d*x+c)-1))^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.46 \[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=\left [\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \sqrt {-\frac {1}{a}} - {\left (3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) - 2 \, \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right )}{2 \, a d}, \frac {\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right )}{a d}\right ] \]
[1/2*(sqrt(2)*a*sqrt(-1/a)*log(-(2*sqrt(2)*(cos(d*x + c)^2 + cos(d*x + c)) *sqrt((a*cos(d*x + c) - a)/cos(d*x + c))*sqrt(-1/a) - (3*cos(d*x + c) + 1) *sin(d*x + c))/((cos(d*x + c) - 1)*sin(d*x + c))) - 2*sqrt(-a)*log((2*(cos (d*x + c)^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c )) - (2*a*cos(d*x + c) + a)*sin(d*x + c))/sin(d*x + c)))/(a*d), (sqrt(2)*s qrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c))*cos(d*x + c) /(sqrt(a)*sin(d*x + c))) - 2*sqrt(a)*arctan(sqrt((a*cos(d*x + c) - a)/cos( d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))))/(a*d)]
\[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx \]
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 698, normalized size of antiderivative = 8.02 \[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} \sqrt {a} \arctan \left (\frac {{\left ({\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{4} + 16 \, \cos \left (d x + c\right )^{4} + 16 \, \sin \left (d x + c\right )^{4} + 8 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} {\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2} + 64 \, \cos \left (d x + c\right )^{3} + 32 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right )^{2} + 64 \, \cos \left (d x + c\right ) + 16\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\frac {8 \, {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2}}, \frac {{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2} + 4 \, \cos \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )^{2} + 8 \, \cos \left (d x + c\right ) + 4}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2}}\right )\right ) + 2 \, \sin \left (d x + c\right )}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}}, \frac {{\left ({\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{4} + 16 \, \cos \left (d x + c\right )^{4} + 16 \, \sin \left (d x + c\right )^{4} + 8 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} {\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2} + 64 \, \cos \left (d x + c\right )^{3} + 32 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right )^{2} + 64 \, \cos \left (d x + c\right ) + 16\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\frac {8 \, {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2}}, \frac {{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2} + 4 \, \cos \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )^{2} + 8 \, \cos \left (d x + c\right ) + 4}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}^{2}}\right )\right ) + 2 \, \cos \left (d x + c\right ) + 2}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} - 2 \right |}}\right ) - \sqrt {a} \arctan \left ({\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + \sin \left (d x + c\right ), {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + \cos \left (d x + c\right )\right )}{a d} \]
-(sqrt(2)*sqrt(a)*arctan2(((abs(2*e^(I*d*x + I*c) - 2)^4 + 16*cos(d*x + c) ^4 + 16*sin(d*x + c)^4 + 8*(cos(d*x + c)^2 - sin(d*x + c)^2 + 2*cos(d*x + c) + 1)*abs(2*e^(I*d*x + I*c) - 2)^2 + 64*cos(d*x + c)^3 + 32*(cos(d*x + c )^2 + 2*cos(d*x + c) + 1)*sin(d*x + c)^2 + 96*cos(d*x + c)^2 + 64*cos(d*x + c) + 16)^(1/4)*sin(1/2*arctan2(8*(cos(d*x + c) + 1)*sin(d*x + c)/abs(2*e ^(I*d*x + I*c) - 2)^2, (abs(2*e^(I*d*x + I*c) - 2)^2 + 4*cos(d*x + c)^2 - 4*sin(d*x + c)^2 + 8*cos(d*x + c) + 4)/abs(2*e^(I*d*x + I*c) - 2)^2)) + 2* sin(d*x + c))/abs(2*e^(I*d*x + I*c) - 2), ((abs(2*e^(I*d*x + I*c) - 2)^4 + 16*cos(d*x + c)^4 + 16*sin(d*x + c)^4 + 8*(cos(d*x + c)^2 - sin(d*x + c)^ 2 + 2*cos(d*x + c) + 1)*abs(2*e^(I*d*x + I*c) - 2)^2 + 64*cos(d*x + c)^3 + 32*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sin(d*x + c)^2 + 96*cos(d*x + c) ^2 + 64*cos(d*x + c) + 16)^(1/4)*cos(1/2*arctan2(8*(cos(d*x + c) + 1)*sin( d*x + c)/abs(2*e^(I*d*x + I*c) - 2)^2, (abs(2*e^(I*d*x + I*c) - 2)^2 + 4*c os(d*x + c)^2 - 4*sin(d*x + c)^2 + 8*cos(d*x + c) + 4)/abs(2*e^(I*d*x + I* c) - 2)^2)) + 2*cos(d*x + c) + 2)/abs(2*e^(I*d*x + I*c) - 2)) - sqrt(a)*ar ctan2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^( 1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + sin(d*x + c), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/ 4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + cos(d*x + c) ))/(a*d)
Time = 0.57 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=\frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {2 \, \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{\sqrt {a}}}{d} \]
(sqrt(2)*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/sqrt(a) - 2*ar ctan(1/2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/sqrt(a))/d
Timed out. \[ \int \frac {1}{\sqrt {a-a \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {a-\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]