Integrand size = 25, antiderivative size = 123 \[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=-\frac {\sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {-2-3 \sec (c+d x)}}{d \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {3 \sqrt {3+2 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-2-3 \sec (c+d x)}} \]
-(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2 *c),2/5*5^(1/2))*5^(1/2)*(-2-3*sec(d*x+c))^(1/2)/d/(3+2*cos(d*x+c))^(1/2)/ sec(d*x+c)^(1/2)-3/5*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip ticF(sin(1/2*d*x+1/2*c),2/5*5^(1/2))*(3+2*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/ 2)/d*5^(1/2)/(-2-3*sec(d*x+c))^(1/2)
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {3+2 \cos (c+d x)} \left (5 E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )-3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {4}{5}\right )\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-2-3 \sec (c+d x)}} \]
(Sqrt[3 + 2*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 4/5] - 3*EllipticF[(c + d*x)/2, 4/5])*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])
Time = 0.93 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4349, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {1}{\sqrt {-3 \sec (c+d x)-2} \sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4349 |
| \(\displaystyle -\frac {1}{2} \int \frac {\sqrt {-3 \sec (c+d x)-2}}{\sqrt {\sec (c+d x)}}dx-\frac {3}{2} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3 \sec (c+d x)-2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} \int \frac {\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3}{2} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle -\frac {3}{2} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx-\frac {\sqrt {-3 \sec (c+d x)-2} \int \sqrt {-2 \cos (c+d x)-3}dx}{2 \sqrt {-2 \cos (c+d x)-3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{2} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx-\frac {\sqrt {-3 \sec (c+d x)-2} \int \sqrt {-2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}dx}{2 \sqrt {-2 \cos (c+d x)-3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {3}{2} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} \int \sqrt {\frac {2}{5} \cos (c+d x)+\frac {3}{5}}dx}{2 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{2} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} \int \sqrt {\frac {2}{5} \sin \left (c+d x+\frac {\pi }{2}\right )+\frac {3}{5}}dx}{2 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {3}{2} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle -\frac {3 \sqrt {-2 \cos (c+d x)-3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-2 \cos (c+d x)-3}}dx}{2 \sqrt {-3 \sec (c+d x)-2}}-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt {-2 \cos (c+d x)-3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}}dx}{2 \sqrt {-3 \sec (c+d x)-2}}-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {3 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\frac {2}{5} \cos (c+d x)+\frac {3}{5}}}dx}{2 \sqrt {5} \sqrt {-3 \sec (c+d x)-2}}-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\frac {2}{5} \sin \left (c+d x+\frac {\pi }{2}\right )+\frac {3}{5}}}dx}{2 \sqrt {5} \sqrt {-3 \sec (c+d x)-2}}-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {3 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {4}{5}\right )}{\sqrt {5} d \sqrt {-3 \sec (c+d x)-2}}-\frac {\sqrt {5} \sqrt {-3 \sec (c+d x)-2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}\) |
-((Sqrt[5]*EllipticE[(c + d*x)/2, 4/5]*Sqrt[-2 - 3*Sec[c + d*x]])/(d*Sqrt[ 3 + 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])) - (3*Sqrt[3 + 2*Cos[c + d*x]]*Ell ipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])
3.7.72.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)]), x_Symbol] :> Simp[1/a Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Cs c[e + f*x]], x], x] - Simp[b/(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Cs c[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 7.49 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.67
| method | result | size |
| default | \(\frac {\sqrt {\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+5}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (2 i \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \frac {\sqrt {5}}{5}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \sqrt {5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+25}-5 i \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \frac {\sqrt {5}}{5}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \sqrt {5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+25}-5 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+25 \cot \left (d x +c \right )-25 \csc \left (d x +c \right )\right )}{5 d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+5\right ) \sqrt {-\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(329\) |
| risch | \(-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right ) \sqrt {2}}{d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {-\frac {2 \left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (\frac {-2 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}-2}{\sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (-2 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}-2\right )}}+\frac {2 \left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}\, \sqrt {-5 \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\sqrt {5}}{2}+\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {{\mathrm e}^{i \left (d x +c \right )}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}\, \left (-\sqrt {5}\, \operatorname {EllipticE}\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}\right )+\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}\right )\right )}{5 \sqrt {-2 \,{\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {-2 \left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right ) {\mathrm e}^{i \left (d x +c \right )}}}{d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {-\frac {2 \left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(547\) |
1/5/d*(((1-cos(d*x+c))^2*csc(d*x+c)^2+5)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1) )^(1/2)*(2*I*EllipticF(I*(-cot(d*x+c)+csc(d*x+c)),1/5*5^(1/2))*((1-cos(d*x +c))^2*csc(d*x+c)^2+1)^(1/2)*(5*(1-cos(d*x+c))^2*csc(d*x+c)^2+25)^(1/2)-5* I*EllipticE(I*(-cot(d*x+c)+csc(d*x+c)),1/5*5^(1/2))*((1-cos(d*x+c))^2*csc( d*x+c)^2+1)^(1/2)*(5*(1-cos(d*x+c))^2*csc(d*x+c)^2+25)^(1/2)-5*(1-cos(d*x+ c))^3*csc(d*x+c)^3+25*cot(d*x+c)-25*csc(d*x+c))/((1-cos(d*x+c))^2*csc(d*x+ c)^2+5)/(-((1-cos(d*x+c))^2*csc(d*x+c)^2+1)/((1-cos(d*x+c))^2*csc(d*x+c)^2 -1))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {{\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) + {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right ) + {\rm weierstrassZeta}\left (8, -4, {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right )\right ) + {\rm weierstrassZeta}\left (8, -4, {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right )\right )}{d} \]
(weierstrassPInverse(8, -4, cos(d*x + c) + I*sin(d*x + c) + 1) + weierstra ssPInverse(8, -4, cos(d*x + c) - I*sin(d*x + c) + 1) + weierstrassZeta(8, -4, weierstrassPInverse(8, -4, cos(d*x + c) + I*sin(d*x + c) + 1)) + weier strassZeta(8, -4, weierstrassPInverse(8, -4, cos(d*x + c) - I*sin(d*x + c) + 1)))/d
\[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {- 3 \sec {\left (c + d x \right )} - 2} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \sec \left (d x + c\right ) - 2} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \sec \left (d x + c\right ) - 2} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {-2-3 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\frac {3}{\cos \left (c+d\,x\right )}-2}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]