Integrand size = 27, antiderivative size = 183 \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt [4]{-a+b} \sqrt {c \cos (e+f x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b} \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}}{\sqrt [4]{-a+b}}\right ),-1\right ) \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}}}{\sqrt [4]{a+b} c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}} \]
2*(-a+b)^(1/4)*EllipticF((a+b)^(1/4)*((1+cos(f*x+e)+sin(f*x+e))/(1+cos(f*x +e)-sin(f*x+e)))^(1/2)/(-a+b)^(1/4),I)*2^(1/2)*(c*cos(f*x+e))^(1/2)*((a+b* sin(f*x+e))/(a-b)/(1-sin(f*x+e)))^(1/2)/(a+b)^(1/4)/c/f/((1+cos(f*x+e)+sin (f*x+e))/(1+cos(f*x+e)-sin(f*x+e)))^(1/2)/(a+b*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 c \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {2 (a+b \sin (e+f x))}{(a-b) (-1+\sin (e+f x))}\right ) (-1+\sin (e+f x)) \left (\frac {(a+b) (1+\sin (e+f x))}{(a-b) (-1+\sin (e+f x))}\right )^{3/4} \sqrt {a+b \sin (e+f x)}}{(a+b) f (c \cos (e+f x))^{3/2}} \]
(-2*c*Hypergeometric2F1[1/2, 3/4, 3/2, (-2*(a + b*Sin[e + f*x]))/((a - b)* (-1 + Sin[e + f*x]))]*(-1 + Sin[e + f*x])*(((a + b)*(1 + Sin[e + f*x]))/(( a - b)*(-1 + Sin[e + f*x])))^(3/4)*Sqrt[a + b*Sin[e + f*x]])/((a + b)*f*(c *Cos[e + f*x])^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(393\) vs. \(2(183)=366\).
Time = 0.44 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.15, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3176, 761}
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.
| \(\displaystyle \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3176 |
| \(\displaystyle \frac {2 \sqrt {2} \sqrt {c \cos (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \int \frac {1}{\sqrt {\frac {(a+b) (\cos (e+f x)+\sin (e+f x)+1)^2}{(a-b) (\cos (e+f x)-\sin (e+f x)+1)^2}+1}}d\sqrt {\frac {\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{c f \sqrt {\frac {\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {c \cos (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \left (\frac {\sqrt {a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt {a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right ) \sqrt {\frac {\frac {(a+b) (\sin (e+f x)+\cos (e+f x)+1)^2}{(a-b) (-\sin (e+f x)+\cos (e+f x)+1)^2}+1}{\left (\frac {\sqrt {a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt {a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a+b} \sqrt {\frac {\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt [4]{a-b}}\right ),\frac {1}{2}\right )}{c f \sqrt [4]{a+b} \sqrt {\frac {\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {(a+b) (\sin (e+f x)+\cos (e+f x)+1)^2}{(a-b) (-\sin (e+f x)+\cos (e+f x)+1)^2}+1}}\) |
(Sqrt[2]*(a - b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[2*ArcTan[((a + b)^(1 /4)*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x ])])/(a - b)^(1/4)], 1/2]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]*(1 + (Sqrt[a + b]*(1 + Cos[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]* (1 + Cos[e + f*x] - Sin[e + f*x])))*Sqrt[(1 + ((a + b)*(1 + Cos[e + f*x] + Sin[e + f*x])^2)/((a - b)*(1 + Cos[e + f*x] - Sin[e + f*x])^2))/(1 + (Sqr t[a + b]*(1 + Cos[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x])))^2])/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[1 + ((a + b)*(1 + Cos[e + f*x] + Sin[e + f*x])^2)/((a - b)*(1 + Cos[e + f*x] - Sin[e + f*x])^2)])
3.7.15.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[2*Sqrt[2]*Sqrt[g*Cos[e + f*x]]*(Sqrt[(a + b* Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]/(f*g*Sqrt[a + b*Sin[e + f*x]]*S qrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])) Subst[Int[1/Sqrt[1 + (a + b)*(x^4/(a - b))], x], x, Sqrt[(1 + Cos[e + f*x ] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]], x] /; FreeQ[{a, b, e , f, g}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(642\) vs. \(2(163)=326\).
Time = 10.40 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.51
| method | result | size |
| default | \(-\frac {4 \left (a \sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-a^{2} \sin \left (f x +e \right )+a b \sin \left (f x +e \right )-\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, a +\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, b -\cos \left (f x +e \right ) a^{2}+\cos \left (f x +e \right ) b^{2}+b \sqrt {-a^{2}+b^{2}}-a b +b^{2}\right ) \sqrt {\frac {\left (-a +b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-b \sin \left (f x +e \right )+a \cos \left (f x +e \right )-a \right )}{\left (a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-a \cos \left (f x +e \right )+a \right )}}\, \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \left (-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right ) a}{\left (a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-a \cos \left (f x +e \right )+a \right )}}\, \sqrt {-\frac {\sqrt {-a^{2}+b^{2}}\, \left (\cos \left (f x +e \right )-1+\sin \left (f x +e \right )\right ) a}{\left (-a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-a \cos \left (f x +e \right )+a \right )}}\, F\left (\sqrt {-\frac {\left (a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b \right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}{\left (a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}}, \sqrt {\frac {\left (b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}{\left (-a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}}\right )}{f \sqrt {c \cos \left (f x +e \right )}\, \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {-a^{2}+b^{2}}\, \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}\) | \(643\) |
-4/f*(a*(-a^2+b^2)^(1/2)*sin(f*x+e)-a^2*sin(f*x+e)+a*b*sin(f*x+e)-cos(f*x+ e)*(-a^2+b^2)^(1/2)*a+cos(f*x+e)*(-a^2+b^2)^(1/2)*b-cos(f*x+e)*a^2+cos(f*x +e)*b^2+b*(-a^2+b^2)^(1/2)-a*b+b^2)*((-a+b+(-a^2+b^2)^(1/2))*((-a^2+b^2)^( 1/2)*sin(f*x+e)-b*sin(f*x+e)+a*cos(f*x+e)-a)/(a-b+(-a^2+b^2)^(1/2))/((-a^2 +b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-a*cos(f*x+e)+a))^(1/2)*((-a^2+b^2)^(1/ 2)*(-cos(f*x+e)+1+sin(f*x+e))*a/(a-b+(-a^2+b^2)^(1/2))/((-a^2+b^2)^(1/2)*s in(f*x+e)+b*sin(f*x+e)-a*cos(f*x+e)+a))^(1/2)*(-(-a^2+b^2)^(1/2)*(cos(f*x+ e)-1+sin(f*x+e))*a/(-a-b+(-a^2+b^2)^(1/2))/((-a^2+b^2)^(1/2)*sin(f*x+e)+b* sin(f*x+e)-a*cos(f*x+e)+a))^(1/2)*EllipticF((-(a*csc(f*x+e)-a*cot(f*x+e)-( -a^2+b^2)^(1/2)+b)*(-a+b+(-a^2+b^2)^(1/2))/(a*csc(f*x+e)-a*cot(f*x+e)+(-a^ 2+b^2)^(1/2)+b)/(a-b+(-a^2+b^2)^(1/2)))^(1/2),((b+(-a^2+b^2)^(1/2)+a)*(a-b +(-a^2+b^2)^(1/2))/(-a-b+(-a^2+b^2)^(1/2))/(-a+b+(-a^2+b^2)^(1/2)))^(1/2)) /(c*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)/(-a^2+b^2)^(1/2)/(-a+b+(-a^2+ b^2)^(1/2))
\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {c \cos \left (f x + e\right )} \sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]
integral(sqrt(c*cos(f*x + e))*sqrt(b*sin(f*x + e) + a)/(b*c*cos(f*x + e)*s in(f*x + e) + a*c*cos(f*x + e)), x)
\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {c \cos {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {c \cos \left (f x + e\right )} \sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {c \cos \left (f x + e\right )} \sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {c\,\cos \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \]