Integrand size = 25, antiderivative size = 50 \[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}} \] Output:
2*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*(a*sin(f*x+e))^(1/2)/f/cos(f*x+e)^ (1/2)/(b*tan(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\sin ^2(e+f x)\right ) \sqrt {a \sin (e+f x)} \sin (2 (e+f x))}{2 f \cos ^2(e+f x)^{3/4} \sqrt {b \tan (e+f x)}} \] Input:
Integrate[Sqrt[a*Sin[e + f*x]]/Sqrt[b*Tan[e + f*x]],x]
Output:
(Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2]*Sqrt[a*Sin[e + f*x]]*Sin [2*(e + f*x)])/(2*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[b*Tan[e + f*x]])
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3081, 3042, 3119}
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.
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 3081 |
| \(\displaystyle \frac {\sqrt {a \sin (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a \sin (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}\) |
Input:
Int[Sqrt[a*Sin[e + f*x]]/Sqrt[b*Tan[e + f*x]],x]
Output:
(2*EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[Cos[e + f*x]]*S qrt[b*Tan[e + f*x]])
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ n) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 1)]) || IntegersQ[m - 1/2, n - 1/2])
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.74
| method | result | size |
| default | \(-\frac {2 \sqrt {a \sin \left (f x +e \right )}\, \left (i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (2+\cos \left (f x +e \right )+\sec \left (f x +e \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticE}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (-\cos \left (f x +e \right )-2-\sec \left (f x +e \right )\right )-\sin \left (f x +e \right )\right )}{f \left (1+\cos \left (f x +e \right )\right ) \sqrt {b \tan \left (f x +e \right )}}\) | \(187\) |
| risch | \(-\frac {i \sqrt {2}\, \sqrt {-i a \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) {\mathrm e}^{-i \left (f x +e \right )}}}{f \sqrt {-\frac {i b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {i \left (-\frac {2 \left (a b \,{\mathrm e}^{2 i \left (f x +e \right )}+a b \right )}{a b \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (a b \,{\mathrm e}^{2 i \left (f x +e \right )}+a b \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a b \,{\mathrm e}^{3 i \left (f x +e \right )}+a \,{\mathrm e}^{i \left (f x +e \right )} b}}\right ) \sqrt {2}\, \sqrt {-i a \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) {\mathrm e}^{-i \left (f x +e \right )}}\, \sqrt {a \,{\mathrm e}^{i \left (f x +e \right )} b \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}}{f \sqrt {-\frac {i b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(383\) |
Input:
int((a*sin(f*x+e))^(1/2)/(b*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/f*(a*sin(f*x+e))^(1/2)/(1+cos(f*x+e))/(b*tan(f*x+e))^(1/2)*(I*(1/(1+cos (f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)- cot(f*x+e)),I)*(2+cos(f*x+e)+sec(f*x+e))+I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f *x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(-cos(f *x+e)-2-sec(f*x+e))-sin(f*x+e))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} \sqrt {-a b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {\frac {1}{2}} \sqrt {-a b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )\right )}}{b f} \] Input:
integrate((a*sin(f*x+e))^(1/2)/(b*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2*(sqrt(1/2)*sqrt(-a*b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + sqrt(1/2)*sqrt(-a*b)*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(b*f)
\[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\int \frac {\sqrt {a \sin {\left (e + f x \right )}}}{\sqrt {b \tan {\left (e + f x \right )}}}\, dx \] Input:
integrate((a*sin(f*x+e))**(1/2)/(b*tan(f*x+e))**(1/2),x)
Output:
Integral(sqrt(a*sin(e + f*x))/sqrt(b*tan(e + f*x)), x)
\[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right )}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \] Input:
integrate((a*sin(f*x+e))^(1/2)/(b*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*sin(f*x + e))/sqrt(b*tan(f*x + e)), x)
\[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right )}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \] Input:
integrate((a*sin(f*x+e))^(1/2)/(b*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*sin(f*x + e))/sqrt(b*tan(f*x + e)), x)
Timed out. \[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\int \frac {\sqrt {a\,\sin \left (e+f\,x\right )}}{\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \] Input:
int((a*sin(e + f*x))^(1/2)/(b*tan(e + f*x))^(1/2),x)
Output:
int((a*sin(e + f*x))^(1/2)/(b*tan(e + f*x))^(1/2), x)
\[ \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \left (\int \frac {\sqrt {\tan \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right )}{b} \] Input:
int((a*sin(f*x+e))^(1/2)/(b*tan(f*x+e))^(1/2),x)
Output:
(sqrt(b)*sqrt(a)*int((sqrt(tan(e + f*x))*sqrt(sin(e + f*x)))/tan(e + f*x), x))/b