Integrand size = 25, antiderivative size = 208 \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {2 \sqrt {a+b} d \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a f} \]
2*d*cot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^ (1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b) )^(1/2)/b/f-2*c*cot(f*x+e)*EllipticPi((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),( a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b *(1+sec(f*x+e))/(a-b))^(1/2)/a/f
Time = 3.78 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.70 \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left ((-c+d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )+2 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sec (e+f x)}{f \sqrt {a+b \sec (e+f x)}} \]
(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Co s[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((-c + d)*EllipticF[ArcSin[Tan[( e + f*x)/2]], (a - b)/(a + b)] + 2*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2 ]], (a - b)/(a + b)])*Sec[e + f*x])/(f*Sqrt[a + b*Sec[e + f*x]])
Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4409, 3042, 4271, 4319}
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.
| \(\displaystyle \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle c \int \frac {1}{\sqrt {a+b \sec (e+f x)}}dx+d \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \int \frac {1}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a f}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {2 d \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b f}-\frac {2 c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a f}\) |
(2*Sqrt[a + b]*d*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sq rt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-(( b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*Ell ipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/( a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x])) /(a - b))])/(a*f)
3.3.3.3.1 Defintions of rubi rules used
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Time = 17.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {2 \left (\cos \left (f x +e \right )+1\right ) \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) c -\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) d -2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right ) c \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}\) | \(182\) |
| parts | \(\frac {2 c \left (\cos \left (f x +e \right )+1\right ) \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )-2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}-\frac {2 d \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}\) | \(263\) |
2/f*(cos(f*x+e)+1)*(EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*c -EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*d-2*EllipticPi(cot(f *x+e)-csc(f*x+e),-1,((a-b)/(a+b))^(1/2))*c)*(1/(a+b)*(b+a*cos(f*x+e))/(cos (f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(a+b*sec(f*x+e))^(1/2) /(b+a*cos(f*x+e))
Timed out. \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\int \frac {c + d \sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\int \frac {c+\frac {d}{\cos \left (e+f\,x\right )}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]