Integrand size = 33, antiderivative size = 209 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=-\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) c f}+\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{(a-b) c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \]
-2*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)/(a-b)/c/f+EllipticE(tan(f*x+e)/(1+sec(f*x+e)),((a-b)/(a+b))^(1/2))* (1/(1+sec(f*x+e)))^(1/2)*(a+b*sec(f*x+e))^(1/2)/(a-b)/c/f/((a+b*sec(f*x+e) )/(a+b)/(1+sec(f*x+e)))^(1/2)
Time = 13.34 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.79 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (b+a \cos (e+f x)) \sec ^2(e+f x) \left (\frac {2 \sin (e+f x)}{-a+b}-\frac {2 \tan \left (\frac {1}{2} (e+f x)\right )}{-a+b}\right )}{f \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))}-\frac {2 \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^{\frac {3}{2}}(e+f x) \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)} \left ((a-b) E\left (\arcsin \left (\sqrt {\frac {a-b}{a+b}} \tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {(b+a \cos (e+f x)) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+\sqrt {2} \sqrt {\frac {a-b}{a+b}} \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} (b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{\left (\frac {a-b}{a+b}\right )^{3/2} (a+b) f \sqrt {\cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )} \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \]
(Cos[e/2 + (f*x)/2]^2*(b + a*Cos[e + f*x])*Sec[e + f*x]^2*((2*Sin[e + f*x] )/(-a + b) - (2*Tan[(e + f*x)/2])/(-a + b)))/(f*Sqrt[a + b*Sec[e + f*x]]*( c + c*Sec[e + f*x])) - (2*Cos[e/2 + (f*x)/2]^2*Sec[e + f*x]^(3/2)*Sqrt[Cos [(e + f*x)/2]^2*Sec[e + f*x]]*((a - b)*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)] + Sqrt[2]*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + Tan[(e + f* x)/2]^2))/(((a - b)/(a + b))^(3/2)*(a + b)*f*Sqrt[Cos[e + f*x]*Sec[(e + f* x)/2]^4]*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x]))
Time = 0.73 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3042, 4460, 3042, 4319, 4456}
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.
| \(\displaystyle \int \frac {\sec (e+f x)}{(c \sec (e+f x)+c) \sqrt {a+b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (c \csc \left (e+f x+\frac {\pi }{2}\right )+c\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4460 |
| \(\displaystyle \frac {\int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sec (e+f x) c+c}dx}{a-b}-\frac {b \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx}{c (a-b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}{\csc \left (e+f x+\frac {\pi }{2}\right ) c+c}dx}{a-b}-\frac {b \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{c (a-b)}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}{\csc \left (e+f x+\frac {\pi }{2}\right ) c+c}dx}{a-b}-\frac {2 \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{c f (a-b)}\) |
| \(\Big \downarrow \) 4456 |
| \(\displaystyle \frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac {2 \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{c f (a-b)}\) |
(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqr t[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 - b)*c*f) + (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*S qrt[a + b*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*( 1 + Sec[e + f*x]))])
3.3.75.3.1 Defintions of rubi rules used
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_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(-Sqrt[a + b*Csc[e + f*x]])*(Sqrt[c/(c + d*Csc[e + f*x])]/(d*f*Sqrt[c*d*((a + b*Csc[e + f*x])/ ((b*c + a*d)*(c + d*Csc[e + f*x])))]))*EllipticE[ArcSin[c*(Cot[e + f*x]/(c + d*Csc[e + f*x]))], -(b*c - a*d)/(b*c + a*d)], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[b/(b*c - a*d) Int [Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[d/(b*c - a*d) Int[C sc[e + f*x]*(Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x])), x], x] /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[ c^2 - d^2, 0])
Time = 5.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\left (\cos \left (f x +e \right )+1\right ) \left (2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b -a \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )-b \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {a +b \sec \left (f x +e \right )}}{c f \left (a -b \right ) \left (b +a \cos \left (f x +e \right )\right )}\) | \(191\) |
1/c/f/(a-b)*(cos(f*x+e)+1)*(2*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b) )^(1/2))*b-a*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))-b*Ellipt icE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2)))*(cos(f*x+e)/(cos(f*x+e)+1) )^(1/2)*(1/(a+b)*(b+a*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))
\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
integral(sqrt(b*sec(f*x + e) + a)*sec(f*x + e)/(b*c*sec(f*x + e)^2 + (a + b)*c*sec(f*x + e) + a*c), x)
\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {a + b \sec {\left (e + f x \right )}}}\, dx}{c} \]
Integral(sec(e + f*x)/(sqrt(a + b*sec(e + f*x))*sec(e + f*x) + sqrt(a + b* sec(e + f*x))), x)/c
\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]