Integrand size = 25, antiderivative size = 115 \[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=-\frac {2 E\left (\left .\frac {1}{2} (c+\pi +d x)\right |6\right ) \sqrt {-3-2 \sec (c+d x)}}{3 d \sqrt {-2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {4 \sqrt {-2-3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),6\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {-3-2 \sec (c+d x)}} \] Output:
-2/3*EllipticE(cos(1/2*d*x+1/2*c),6^(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)-4/3*(-2-3*cos(d*x+c))^(1/2)*InverseJac obiAM(1/2*d*x+1/2*Pi+1/2*c,6^(1/2))*sec(d*x+c)^(1/2)/d/(-3-2*sec(d*x+c))^( 1/2)
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {2 \sqrt {2+3 \cos (c+d x)} \left (5 E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )-2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {6}{5}\right )\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-3-2 \sec (c+d x)}} \] Input:
Integrate[1/(Sqrt[-3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]
Output:
(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 = 0.73 (sec) , antiderivative size = 115, 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, 3133, 4345, 3042, 3141}
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.
| \(\displaystyle \int \frac {1}{\sqrt {-2 \sec (c+d x)-3} \sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {-2 \csc \left (c+d x+\frac {\pi }{2}\right )-3} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-2 \csc \left (c+d x+\frac {\pi }{2}\right )-3}}dx-\frac {\sqrt {-2 \sec (c+d x)-3} \int \sqrt {-3 \cos (c+d x)-2}dx}{3 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-2 \csc \left (c+d x+\frac {\pi }{2}\right )-3}}dx-\frac {\sqrt {-2 \sec (c+d x)-3} \int \sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}dx}{3 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3133 |
| \(\displaystyle -\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-2 \csc \left (c+d x+\frac {\pi }{2}\right )-3}}dx-\frac {2 \sqrt {-2 \sec (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle -\frac {2 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-3 \cos (c+d x)-2}}dx}{3 \sqrt {-2 \sec (c+d x)-3}}-\frac {2 \sqrt {-2 \sec (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}}dx}{3 \sqrt {-2 \sec (c+d x)-3}}-\frac {2 \sqrt {-2 \sec (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3141 |
| \(\displaystyle -\frac {4 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),6\right )}{3 d \sqrt {-2 \sec (c+d x)-3}}-\frac {2 \sqrt {-2 \sec (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}}\) |
Input:
Int[1/(Sqrt[-3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]
Output:
(-2*EllipticE[(c + Pi + d*x)/2, 6]*Sqrt[-3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*Sqrt[-2 - 3*Cos[c + d*x]]*Elli pticF[(c + Pi + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[-3 - 2*Sec[c + d* x]])
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 = 3.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.98
| method | result | size |
| default | \(-\frac {2 \sqrt {-3-2 \sec \left (d x +c \right )}\, \left (3 i \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-5 i \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+5 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-25 \csc \left (d x +c \right )+25 \cot \left (d x +c \right )\right )}{15 d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-5\right ) \sqrt {\sec \left (d x +c \right )}}\) | \(228\) |
| 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 \,{\mathrm e}^{2 i \left (d x +c \right )}-\frac {8 \,{\mathrm e}^{i \left (d x +c \right )}}{3}-2}{\sqrt {\left (-3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}-3\right ) {\mathrm e}^{i \left (d x +c \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}}}\) | \(576\) |
Input:
int(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/15/d*(-3-2*sec(d*x+c))^(1/2)*(3*I*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),1 /5*I*5^(1/2))*2^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/ (1+cos(d*x+c)))^(1/2)-5*I*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),1/5*I*5^(1/2 ))*2^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+ c)))^(1/2)+5*(1-cos(d*x+c))^3*csc(d*x+c)^3-25*csc(d*x+c)+25*cot(d*x+c))/(( 1-cos(d*x+c))^2*csc(d*x+c)^2-5)/sec(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + 4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, \frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )\right ) + 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, \frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )\right )}{27 \, d} \] Input:
integrate(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="fricas" )
Output:
1/27*(4*sqrt(6)*weierstrassPInverse(-44/27, 784/729, cos(d*x + c) + I*sin( d*x + c) + 4/9) + 4*sqrt(6)*weierstrassPInverse(-44/27, 784/729, cos(d*x + c) - I*sin(d*x + c) + 4/9) + 9*sqrt(6)*weierstrassZeta(-44/27, 784/729, w eierstrassPInverse(-44/27, 784/729, cos(d*x + c) + I*sin(d*x + c) + 4/9)) + 9*sqrt(6)*weierstrassZeta(-44/27, 784/729, weierstrassPInverse(-44/27, 7 84/729, cos(d*x + c) - I*sin(d*x + c) + 4/9)))/d
\[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {- 2 \sec {\left (c + d x \right )} - 3} \sqrt {\sec {\left (c + d x \right )}}}\, dx \] Input:
integrate(1/(-3-2*sec(d*x+c))**(1/2)/sec(d*x+c)**(1/2),x)
Output:
Integral(1/(sqrt(-2*sec(c + d*x) - 3)*sqrt(sec(c + d*x))), x)
\[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="maxima" )
Output:
integrate(1/(sqrt(-2*sec(d*x + c) - 3)*sqrt(sec(d*x + c))), x)
\[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-2*sec(d*x + c) - 3)*sqrt(sec(d*x + c))), x)
Timed out. \[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\frac {2}{\cos \left (c+d\,x\right )}-3}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(1/((- 2/cos(c + d*x) - 3)^(1/2)*(1/cos(c + d*x))^(1/2)),x)
Output:
int(1/((- 2/cos(c + d*x) - 3)^(1/2)*(1/cos(c + d*x))^(1/2)), x)
\[ \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx=-\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {-2 \sec \left (d x +c \right )-3}}{2 \sec \left (d x +c \right )^{2}+3 \sec \left (d x +c \right )}d x \right ) \] Input:
int(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x)
Output:
- int((sqrt(sec(c + d*x))*sqrt( - 2*sec(c + d*x) - 3))/(2*sec(c + d*x)**2 + 3*sec(c + d*x)),x)