Integrand size = 25, antiderivative size = 127 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=-\frac {4 \sqrt {2+3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {3+2 \sec (c+d x)}}+\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {3+2 \sec (c+d x)}}{3 d \sqrt {2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]
-4/15*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d* x+1/2*c),1/5*30^(1/2))*(2+3*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/2)/d*5^(1/2)/( 3+2*sec(d*x+c))^(1/2)+2/3*(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),1/5*30^(1/2))*5^(1/2)*(3+2*sec(d*x+c))^(1/2)/ d/(2+3*cos(d*x+c))^(1/2)/sec(d*x+c)^(1/2)
Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=\frac {2 \sqrt {2+3 \cos (c+d x)} \left (5 E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )-2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {6}{5}\right )\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {3+2 \sec (c+d x)}} \]
(2*Sqrt[2 + 3*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 6/5] - 2*EllipticF[( c + d*x)/2, 6/5])*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x] ])
Time = 0.69 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4349, 3042, 4343, 3042, 3132, 4345, 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.
| \(\displaystyle \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {2 \sec (c+d x)+3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {2 \csc \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
| \(\Big \downarrow \) 4349 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt {2 \sec (c+d x)+3}}{\sqrt {\sec (c+d x)}}dx-\frac {2}{3} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {2 \sec (c+d x)+3}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt {2 \csc \left (c+d x+\frac {\pi }{2}\right )+3}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {2 \csc \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {\sqrt {2 \sec (c+d x)+3} \int \sqrt {3 \cos (c+d x)+2}dx}{3 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {2 \csc \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {2 \sec (c+d x)+3} \int \sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}dx}{3 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {2 \csc \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)+3} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{3 d \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {2 \csc \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)+3} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{3 d \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {3 \cos (c+d x)+2}}dx}{3 \sqrt {2 \sec (c+d x)+3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)+3} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{3 d \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}}dx}{3 \sqrt {2 \sec (c+d x)+3}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)+3} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{3 d \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {4 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {6}{5}\right )}{3 \sqrt {5} d \sqrt {2 \sec (c+d x)+3}}\) |
(-4*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6/5]*Sqrt[Sec[c + d*x] ])/(3*Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]]) + (2*Sqrt[5]*EllipticE[(c + d*x) /2, 6/5]*Sqrt[3 + 2*Sec[c + d*x]])/(3*d*Sqrt[2 + 3*Cos[c + d*x]]*Sqrt[Sec[ c + d*x]])
3.7.73.3.1 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[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[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 = 7.48 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.67
| method | result | size |
| default | \(\frac {2 \sqrt {\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-5}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (3 \operatorname {EllipticF}\left (\frac {\sqrt {5}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{5}, i \sqrt {5}\right ) \sqrt {5}\, \sqrt {-5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+25}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}-\operatorname {EllipticE}\left (\frac {\sqrt {5}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{5}, i \sqrt {5}\right ) \sqrt {5}\, \sqrt {-5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+25}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}+5 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+25 \cot \left (d x +c \right )-25 \csc \left (d x +c \right )\right )}{15 d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-5\right ) \sqrt {-\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(339\) |
| risch | \(-\frac {i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right ) \sqrt {2}}{3 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 {3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}{3 \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}}+\frac {\left (\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}+\frac {i \sqrt {5}}{3}}{\frac {2}{3}+\frac {i \sqrt {5}}{3}}}\, \sqrt {30}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{-\frac {2}{3}-\frac {i \sqrt {5}}{3}}}\, \left (-\frac {2 i \sqrt {5}\, \operatorname {EllipticE}\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}+\frac {i \sqrt {5}}{3}}{\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (-\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )}{3}+\left (-\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}+\frac {i \sqrt {5}}{3}}{\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (-\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )\right )}{5 \sqrt {3 \,{\mathrm e}^{3 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\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 {3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(573\) |
2/15/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)*(3*EllipticF(1/5*5^(1/2)*(-cot(d*x+c)+csc(d*x+c)),I*5^(1/2))*5^(1 /2)*(-5*(1-cos(d*x+c))^2*csc(d*x+c)^2+25)^(1/2)*((1-cos(d*x+c))^2*csc(d*x+ c)^2+1)^(1/2)-EllipticE(1/5*5^(1/2)*(-cot(d*x+c)+csc(d*x+c)),I*5^(1/2))*5^ (1/2)*(-5*(1-cos(d*x+c))^2*csc(d*x+c)^2+25)^(1/2)*((1-cos(d*x+c))^2*csc(d* x+c)^2+1)^(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 = 108, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=\frac {4 i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) - 4 i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + 9 i \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, \frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )\right ) - 9 i \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, \frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )\right )}{27 \, d} \]
1/27*(4*I*sqrt(6)*weierstrassPInverse(-44/27, 784/729, cos(d*x + c) + I*si n(d*x + c) + 4/9) - 4*I*sqrt(6)*weierstrassPInverse(-44/27, 784/729, cos(d *x + c) - I*sin(d*x + c) + 4/9) + 9*I*sqrt(6)*weierstrassZeta(-44/27, 784/ 729, weierstrassPInverse(-44/27, 784/729, cos(d*x + c) + I*sin(d*x + c) + 4/9)) - 9*I*sqrt(6)*weierstrassZeta(-44/27, 784/729, weierstrassPInverse(- 44/27, 784/729, cos(d*x + c) - I*sin(d*x + c) + 4/9)))/d
\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 \sec {\left (c + d x \right )} + 3} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \sec \left (d x + c\right ) + 3} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \sec \left (d x + c\right ) + 3} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}+3}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]