Integrand size = 35, antiderivative size = 158 \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f} \]
sec(f*x+e)*(d*sin(f*x+e))^(1/2)*(a+b*sin(f*x+e))^(1/2)/d/f-EllipticF(d^(1/ 2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^ (1/2))*(a+b)^(1/2)*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b)) ^(1/2)*tan(f*x+e)/f/d^(1/2)
Time = 4.45 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {4 a^2 \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {a+b \sin (e+f x)}{a (-1+\sin (e+f x))}}\right ),\frac {2 a}{a-b}\right ) \sec (e+f x) \sqrt {-\frac {(a+b) \sin (e+f x) (a+b \sin (e+f x))}{a^2 (-1+\sin (e+f x))^2}} \sin ^4\left (\frac {1}{4} (2 e-\pi +2 f x)\right )+(a+b) (a+b \sin (e+f x)) \tan (e+f x)}{(a+b) f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \]
(4*a^2*Sqrt[-(((a + b)*Cot[(2*e - Pi + 2*f*x)/4]^2)/(a - b))]*EllipticF[Ar cSin[Sqrt[-((a + b*Sin[e + f*x])/(a*(-1 + Sin[e + f*x])))]], (2*a)/(a - b) ]*Sec[e + f*x]*Sqrt[-(((a + b)*Sin[e + f*x]*(a + b*Sin[e + f*x]))/(a^2*(-1 + Sin[e + f*x])^2))]*Sin[(2*e - Pi + 2*f*x)/4]^4 + (a + b)*(a + b*Sin[e + f*x])*Tan[e + f*x])/((a + b)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f* x]])
Time = 0.57 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3042, 3367, 3042, 3295}
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.
| \(\displaystyle \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{\cos (e+f x)^2 \sqrt {d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3367 |
| \(\displaystyle \frac {1}{2} a \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx+\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx+\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{\sqrt {d} f}\) |
(Sec[e + f*x]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(d*f) - (Sqrt [a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/( a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*S qrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f)
3.15.78.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*(g*Cos [e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^m/(d*f*g*(2*m + 1))), x] + Simp[2*a*(m/(g^2*(2*m + 1))) Int[(g*Cos[e + f*x])^(p + 2)*( (a + b*Sin[e + f*x])^(m - 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(949\) vs. \(2(142)=284\).
Time = 2.59 (sec) , antiderivative size = 950, normalized size of antiderivative = 6.01
| method | result | size |
| default | \(\frac {\left (\sqrt {-a^{2}+b^{2}}\, \sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {-\frac {a \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, \cos \left (f x +e \right )+\sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {-\frac {a \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}\, \sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {-\frac {a \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}+\sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {-\frac {a \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, b -\sqrt {2}\, b \cos \left (f x +e \right )+\tan \left (f x +e \right ) \sqrt {2}\, a +\sec \left (f x +e \right ) \sqrt {2}\, b \right ) \sqrt {2}}{2 f \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\) | \(950\) |
1/2/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*((-a^2+b^2)^(1/2)*(-(a*c sc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+ (-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2 )^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2) *((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*(1/(b+(-a^2+b^2)^(1/2))*(a *csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*cos(f*x+e)+(-(a*csc(f* x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2 +b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/ 2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+ (-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*(1/(b+(-a^2+b^2)^(1/2))*(a*csc( f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*b*cos(f*x+e)+(-a^2+b^2)^(1/ 2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2 )*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+ (-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/ 2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*(1/(b+(-a^2+b^2)^ (1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)+(-(a*csc(f*x+ e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b ^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2) )*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(- a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*(1/(b+(-a^2+b^2)^(1/2))*(a*csc...
\[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {\sqrt {a + b \sin {\left (e + f x \right )}} \sec ^{2}{\left (e + f x \right )}}{\sqrt {d \sin {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{{\cos \left (e+f\,x\right )}^2\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]