Integrand size = 39, antiderivative size = 168 \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a f \sqrt {c+d \sec (e+f x)}}+\frac {2 (a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a (a+b) f \sqrt {c+d \sec (e+f x)}} \]
2*d*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x +1/2*e),2,2^(1/2)*(c/(c+d))^(1/2))*((d+c*cos(f*x+e))/(c+d))^(1/2)*(g*sec(f *x+e))^(1/2)/a/f/(c+d*sec(f*x+e))^(1/2)+2*(a*c-b*d)*(cos(1/2*f*x+1/2*e)^2) ^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(a+b),2^(1/2)* (c/(c+d))^(1/2))*((d+c*cos(f*x+e))/(c+d))^(1/2)*(g*sec(f*x+e))^(1/2)/a/(a+ b)/f/(c+d*sec(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 25.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=-\frac {2 i \sqrt {-\frac {c (-1+\cos (e+f x))}{c+d}} \sqrt {\frac {c (1+\cos (e+f x))}{c-d}} \cot (e+f x) \left (\operatorname {EllipticPi}\left (1-\frac {c}{d},i \text {arcsinh}\left (\sqrt {\frac {1}{c-d}} \sqrt {d+c \cos (e+f x)}\right ),\frac {-c+d}{c+d}\right )-\operatorname {EllipticPi}\left (\frac {b (-c+d)}{-a c+b d},i \text {arcsinh}\left (\sqrt {\frac {1}{c-d}} \sqrt {d+c \cos (e+f x)}\right ),\frac {-c+d}{c+d}\right )\right ) \sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a \sqrt {\frac {1}{c-d}} f \sqrt {d+c \cos (e+f x)}} \]
((-2*I)*Sqrt[-((c*(-1 + Cos[e + f*x]))/(c + d))]*Sqrt[(c*(1 + Cos[e + f*x] ))/(c - d)]*Cot[e + f*x]*(EllipticPi[1 - c/d, I*ArcSinh[Sqrt[(c - d)^(-1)] *Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)] - EllipticPi[(b*(-c + d))/(- (a*c) + b*d), I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)])*Sqrt[g*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(a*Sqrt[(c - d)^(-1)]*f*Sqrt[d + c*Cos[e + f*x]])
Time = 2.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3441, 3042, 4459, 3042, 4346, 3042, 3286, 3042, 3284, 4463, 3042, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}{a+b \sin \left (e+f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3441 |
| \(\displaystyle \frac {\int \frac {(g \sec (e+f x))^{3/2} \sqrt {c+d \sec (e+f x)}}{b+a \sec (e+f x)}dx}{g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2} \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}{b+a \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{g}\) |
| \(\Big \downarrow \) 4459 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {(g \sec (e+f x))^{3/2}}{(b+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}}dx}{a}+\frac {d \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {c+d \sec (e+f x)}}dx}{a}}{g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {d \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}}{g}\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {d g \sqrt {g \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\sec (e+f x)}{\sqrt {d+c \cos (e+f x)}}dx}{a \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {d g \sqrt {g \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {1}{\sin \left (e+f x+\frac {\pi }{2}\right ) \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \int \frac {\sec (e+f x)}{\sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}}dx}{a \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \int \frac {1}{\sin \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\frac {d}{c+d}+\frac {c \sin \left (e+f x+\frac {\pi }{2}\right )}{c+d}}}dx}{a \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {(a c-b d) \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 4463 |
| \(\displaystyle \frac {\frac {g (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {1}{(a+b \cos (e+f x)) \sqrt {d+c \cos (e+f x)}}dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {2 d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {g (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {1}{\left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {2 d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {g (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \int \frac {1}{(a+b \cos (e+f x)) \sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}}dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {2 d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {g (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \int \frac {1}{\left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {\frac {d}{c+d}+\frac {c \sin \left (e+f x+\frac {\pi }{2}\right )}{c+d}}}dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {2 d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}}}{g}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {2 g (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f (a+b) \sqrt {c+d \sec (e+f x)}}+\frac {2 d g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}}}{g}\) |
((2*d*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[2, (e + f*x)/2, (2*c )/(c + d)]*Sqrt[g*Sec[e + f*x]])/(a*f*Sqrt[c + d*Sec[e + f*x]]) + (2*(a*c - b*d)*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*b)/(a + b), (e + f*x)/2, (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/(a*(a + b)*f*Sqrt[c + d*Sec[ e + f*x]]))/g
3.3.72.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp [g^m Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f *x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[((csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[b/d Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/d Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[g*Sqr t[g*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]) Int [1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 8.24 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.63
| method | result | size |
| default | \(\frac {2 i \sqrt {g \sec \left (f x +e \right )}\, \cos \left (f x +e \right ) \sqrt {c +d \sec \left (f x +e \right )}\, \left (\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a^{2} c -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a^{2} d +\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a b c -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a b d +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {c -d}{c +d}}\right ) a^{2} d -2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {c -d}{c +d}}\right ) b^{2} d -2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right ) a b c +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right ) b^{2} d \right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}}{f a \left (a -b \right ) \left (a +b \right ) \left (d +c \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(442\) |
2*I/f/a/(a-b)/(a+b)*(g*sec(f*x+e))^(1/2)*cos(f*x+e)*(c+d*sec(f*x+e))^(1/2) *(EllipticF(I*(cot(f*x+e)-csc(f*x+e)),(-(c-d)/(c+d))^(1/2))*a^2*c-Elliptic F(I*(cot(f*x+e)-csc(f*x+e)),(-(c-d)/(c+d))^(1/2))*a^2*d+EllipticF(I*(cot(f *x+e)-csc(f*x+e)),(-(c-d)/(c+d))^(1/2))*a*b*c-EllipticF(I*(cot(f*x+e)-csc( f*x+e)),(-(c-d)/(c+d))^(1/2))*a*b*d+2*EllipticPi(I*(cot(f*x+e)-csc(f*x+e)) ,-1,I*((c-d)/(c+d))^(1/2))*a^2*d-2*EllipticPi(I*(cot(f*x+e)-csc(f*x+e)),-1 ,I*((c-d)/(c+d))^(1/2))*b^2*d-2*EllipticPi(I*(cot(f*x+e)-csc(f*x+e)),(a-b) /(a+b),I*((c-d)/(c+d))^(1/2))*a*b*c+2*EllipticPi(I*(cot(f*x+e)-csc(f*x+e)) ,(a-b)/(a+b),I*((c-d)/(c+d))^(1/2))*b^2*d)*(1/(c+d)*(d+c*cos(f*x+e))/(cos( f*x+e)+1))^(1/2)/(d+c*cos(f*x+e))/(1/(cos(f*x+e)+1))^(1/2)
Timed out. \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int \frac {\sqrt {g \sec {\left (e + f x \right )}} \sqrt {c + d \sec {\left (e + f x \right )}}}{a + b \cos {\left (e + f x \right )}}\, dx \]
\[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sqrt {g \sec \left (f x + e\right )}}{b \cos \left (f x + e\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sqrt {g \sec \left (f x + e\right )}}{b \cos \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int \frac {\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {g}{\cos \left (e+f\,x\right )}}}{a+b\,\cos \left (e+f\,x\right )} \,d x \]