Integrand size = 37, antiderivative size = 209 \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}} \]
2*EllipticPi((d*cos(f*x+e))^(1/2)/d^(1/2)/(1+sin(f*x+e))^(1/2),-a/(b-(-a^2 +b^2)^(1/2)),I)*2^(1/2)*d^(1/2)*sin(f*x+e)^(1/2)/f/(-a^2+b^2)^(1/2)/(g*sin (f*x+e))^(1/2)-2*EllipticPi((d*cos(f*x+e))^(1/2)/d^(1/2)/(1+sin(f*x+e))^(1 /2),-a/(b+(-a^2+b^2)^(1/2)),I)*2^(1/2)*d^(1/2)*sin(f*x+e)^(1/2)/f/(-a^2+b^ 2)^(1/2)/(g*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 17.92 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.78 \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {d \cos (e+f x)} \left (b+a \sqrt {\sec ^2(e+f x)}\right ) \sqrt {\tan (e+f x)} \left (\frac {\sqrt {a} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )\right )}{4 \sqrt {2} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right ) \sqrt {\tan (e+f x)}}{\sqrt {\sec ^2(e+f x)} \left (a^2-b^2+a^2 \tan ^2(e+f x)\right ) \left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )+2 \left (2 a^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )+\left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )\right ) \tan ^2(e+f x)\right )}\right )}{f (a+b \cos (e+f x)) \sqrt {\sec ^2(e+f x)} \sqrt {g \sin (e+f x)}} \]
(2*Sqrt[d*Cos[e + f*x]]*(b + a*Sqrt[Sec[e + f*x]^2])*Sqrt[Tan[e + f*x]]*(( Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1 /4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4) ] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f *x]] + a*Tan[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2) ^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4) ) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, -((a^2*Ta n[e + f*x]^2)/(a^2 - b^2))]*Sqrt[Tan[e + f*x]])/(Sqrt[Sec[e + f*x]^2]*(a^2 - b^2 + a^2*Tan[e + f*x]^2)*(-5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -T an[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))] + 2*(2*a^2*AppellF1[5/ 4, 1/2, 2, 9/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))] + (a ^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^ 2)/(a^2 - b^2))])*Tan[e + f*x]^2))))/(f*(a + b*Cos[e + f*x])*Sqrt[Sec[e + f*x]^2]*Sqrt[g*Sin[e + f*x]])
Time = 0.83 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {3042, 3387, 3042, 3386, 1542}
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.
| \(\displaystyle \int \frac {\sqrt {d \cos (e+f x)}}{\sqrt {g \sin (e+f x)} (a+b \cos (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-d \sin \left (e+f x-\frac {\pi }{2}\right )}}{\sqrt {g \cos \left (e+f x-\frac {\pi }{2}\right )} \left (a-b \sin \left (e+f x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3387 |
| \(\displaystyle \frac {\sqrt {\sin (e+f x)} \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {\sin (e+f x)}}dx}{\sqrt {g \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (e+f x)} \int \frac {\sqrt {-d \sin \left (e+f x-\frac {\pi }{2}\right )}}{\sqrt {\cos \left (e+f x-\frac {\pi }{2}\right )} \left (a-b \sin \left (e+f x-\frac {\pi }{2}\right )\right )}dx}{\sqrt {g \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3386 |
| \(\displaystyle \frac {\sqrt {\sin (e+f x)} \left (-\frac {2 \sqrt {2} d \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\left (b-\sqrt {b^2-a^2}\right ) d+\frac {a \cos (e+f x) d}{\sin (e+f x)+1}\right )}d\frac {\sqrt {d \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{f}-\frac {2 \sqrt {2} d \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\left (b+\sqrt {b^2-a^2}\right ) d+\frac {a \cos (e+f x) d}{\sin (e+f x)+1}\right )}d\frac {\sqrt {d \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{f}\right )}{\sqrt {g \sin (e+f x)}}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\sqrt {\sin (e+f x)} \left (-\frac {2 \sqrt {2} \sqrt {d} \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{f \left (b-\sqrt {b^2-a^2}\right )}-\frac {2 \sqrt {2} \sqrt {d} \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{f \left (\sqrt {b^2-a^2}+b\right )}\right )}{\sqrt {g \sin (e+f x)}}\) |
(((-2*Sqrt[2]*(1 - b/Sqrt[-a^2 + b^2])*Sqrt[d]*EllipticPi[-(a/(b - Sqrt[-a ^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])] , -1])/((b - Sqrt[-a^2 + b^2])*f) - (2*Sqrt[2]*(1 + b/Sqrt[-a^2 + b^2])*Sq rt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/ (Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1])/((b + Sqrt[-a^2 + b^2])*f))*Sqrt[S in[e + f*x]])/Sqrt[g*Sin[e + f*x]]
3.1.2.3.1 Defintions of rubi rules used
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[2*Sqrt[2]*d*((b + q)/(f*q)) Subst[Int[1/((d*(b + q) + a*x^2)*Sq rt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Simp[2*Sqrt[2]*d*((b - q)/(f*q)) Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; F reeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.) ]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Cos[e + f *x]]/Sqrt[g*Cos[e + f*x]] Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^ 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs. \(2(173)=346\).
Time = 4.77 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.43
| method | result | size |
| default | \(\frac {\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \left (2 \sqrt {-a^{2}+b^{2}}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\cos \left (f x +e \right ) d}\, \left (\sec \left (f x +e \right )+1\right ) \sqrt {2}\, a}{f \sqrt {g \sin \left (f x +e \right )}\, \sqrt {-a^{2}+b^{2}}\, \left (-b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (b +\sqrt {-a^{2}+b^{2}}-a \right )}\) | \(508\) |
1/f*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-cs c(f*x+e)+cot(f*x+e))^(1/2)*(2*(-a^2+b^2)^(1/2)*EllipticF((-cot(f*x+e)+csc( f*x+e)+1)^(1/2),1/2*2^(1/2))+a*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2) ,(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))-EllipticPi((-cot(f*x+e)+csc (f*x+e)+1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b-(-a^2+b^2 )^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+ b))^(1/2)),1/2*2^(1/2))-a*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-(a- b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))+EllipticPi((-cot(f*x+e)+csc(f* x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b-(-a^2+b^2) ^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a +b))^(1/2)),1/2*2^(1/2)))*(cos(f*x+e)*d)^(1/2)/(g*sin(f*x+e))^(1/2)*(sec(f *x+e)+1)*2^(1/2)*a/(-a^2+b^2)^(1/2)/(-b+(-a^2+b^2)^(1/2)+a)/(b+(-a^2+b^2)^ (1/2)-a)
Timed out. \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int \frac {\sqrt {d \cos {\left (e + f x \right )}}}{\sqrt {g \sin {\left (e + f x \right )}} \left (a + b \cos {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \cos \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \cos \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int \frac {\sqrt {d\,\cos \left (e+f\,x\right )}}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \]