Integrand size = 27, antiderivative size = 225 \[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=-\frac {2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {c}{c+d},\arcsin \left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right ),\frac {c-d}{c+d}\right ) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (1+\sec (e+f x))}{c+d \sec (e+f x)}} (c+d \sec (e+f x))}{a \sqrt {c+d} f}-\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \]
-2*cot(f*x+e)*EllipticPi((c+d)^(1/2)/(c+d*sec(f*x+e))^(1/2),c/(c+d),((c-d) /(c+d))^(1/2))*(c+d*sec(f*x+e))*(-d*(1-sec(f*x+e))/(c+d*sec(f*x+e)))^(1/2) *(d*(1+sec(f*x+e))/(c+d*sec(f*x+e)))^(1/2)/a/f/(c+d)^(1/2)-EllipticE(tan(f *x+e)/(1+sec(f*x+e)),((c-d)/(c+d))^(1/2))*(1/(1+sec(f*x+e)))^(1/2)*(c+d*se c(f*x+e))^(1/2)/a/f/((c+d*sec(f*x+e))/(c+d)/(1+sec(f*x+e)))^(1/2)
Time = 7.53 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=-\frac {4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left ((c+d) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+2 (c-d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )-4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c+d) f (1+\cos (e+f x))^2 \sqrt {\frac {d+c \cos (e+f x)}{(c+d) (1+\cos (e+f x))}}} \]
(-4*Cos[(e + f*x)/2]^4*((c + d)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (c - d )/(c + d)] + 2*(c - d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d) ] - 4*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*(c + d)*f*(1 + Cos[e + f*x])^2*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))])
Time = 0.64 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 4413, 3042, 4267, 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.
\(\displaystyle \int \frac {\sqrt {c+d \sec (e+f x)}}{a \sec (e+f x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}{a \csc \left (e+f x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 4413 |
\(\displaystyle \frac {\int \sqrt {c+d \sec (e+f x)}dx}{a}-\int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sec (e+f x) a+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a}-\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}{\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx\) |
\(\Big \downarrow \) 4267 |
\(\displaystyle -\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}{\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx-\frac {2 \cot (e+f x) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \operatorname {EllipticPi}\left (\frac {c}{c+d},\arcsin \left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right ),\frac {c-d}{c+d}\right )}{a f \sqrt {c+d}}\) |
\(\Big \downarrow \) 4456 |
\(\displaystyle -\frac {2 \cot (e+f x) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \operatorname {EllipticPi}\left (\frac {c}{c+d},\arcsin \left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right ),\frac {c-d}{c+d}\right )}{a f \sqrt {c+d}}-\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}\) |
(-2*Cot[e + f*x]*EllipticPi[c/(c + d), ArcSin[Sqrt[c + d]/Sqrt[c + d*Sec[e + f*x]]], (c - d)/(c + d)]*Sqrt[-((d*(1 - Sec[e + f*x]))/(c + d*Sec[e + f *x]))]*Sqrt[(d*(1 + Sec[e + f*x]))/(c + d*Sec[e + f*x])]*(c + d*Sec[e + f* x]))/(a*Sqrt[c + d]*f) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x]) ], (c - d)/(c + d)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]] )/(a*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])
3.2.45.3.1 Defintions of rubi rules used
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*((a + b *Csc[c + d*x])/(d*Rt[a + b, 2]*Cot[c + d*x]))*Sqrt[b*((1 + Csc[c + d*x])/(a + b*Csc[c + d*x]))]*Sqrt[(-b)*((1 - Csc[c + d*x])/(a + b*Csc[c + d*x]))]*E llipticPi[a/(a + b), ArcSin[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b) /(a + b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/(csc[(e_.) + (f_.)*(x_)]*(d_ .) + (c_)), x_Symbol] :> Simp[1/c Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[d/c Int[Csc[e + f*x]*(Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x])) , x], x] /; FreeQ[{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])
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]
Time = 5.05 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\left (\cos \left (f x +e \right )+1\right ) \left (2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) c -2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) d +c \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right )+d \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right )-4 c \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {c -d}{c +d}}\right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {c +d \sec \left (f x +e \right )}}{a f \left (d +c \cos \left (f x +e \right )\right )}\) | \(247\) |
1/a/f*(cos(f*x+e)+1)*(2*EllipticF(cot(f*x+e)-csc(f*x+e),((c-d)/(c+d))^(1/2 ))*c-2*EllipticF(cot(f*x+e)-csc(f*x+e),((c-d)/(c+d))^(1/2))*d+c*EllipticE( cot(f*x+e)-csc(f*x+e),((c-d)/(c+d))^(1/2))+d*EllipticE(cot(f*x+e)-csc(f*x+ e),((c-d)/(c+d))^(1/2))-4*c*EllipticPi(cot(f*x+e)-csc(f*x+e),-1,((c-d)/(c+ d))^(1/2)))*(1/(c+d)*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(c os(f*x+e)+1))^(1/2)*(c+d*sec(f*x+e))^(1/2)/(d+c*cos(f*x+e))
\[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c}}{a \sec \left (f x + e\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=\frac {\int \frac {\sqrt {c + d \sec {\left (e + f x \right )}}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
\[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c}}{a \sec \left (f x + e\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c}}{a \sec \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx=\int \frac {\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}}{a+\frac {a}{\cos \left (e+f\,x\right )}} \,d x \]