Integrand size = 23, antiderivative size = 51 \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\frac {E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \] Output:
-EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*sin(f*x+e)^(1/2)/f/(b*sec(f*x+e))^(1 /2)/sin(2*f*x+2*e)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.82 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}} \] Input:
Integrate[Sqrt[Sin[e + f*x]]/Sqrt[b*Sec[e + f*x]],x]
Output:
-((b*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^( 1/4))/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]]))
Time = 0.57 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3065, 3042, 3052, 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.
\(\displaystyle \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}}dx\) |
\(\Big \downarrow \) 3065 |
\(\displaystyle \frac {\int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)}dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)}dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{\sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{\sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}\) |
Input:
Int[Sqrt[Sin[e + f*x]]/Sqrt[b*Sec[e + f*x]],x]
Output:
(EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]]* Sqrt[Sin[2*e + 2*f*x]])
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && Int egerQ[m - 1/2] && IntegerQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(213\) vs. \(2(47)=94\).
Time = 0.35 (sec) , antiderivative size = 214, normalized size of antiderivative = 4.20
method | result | size |
default | \(-\frac {\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (2+2 \sec \left (f x +e \right )\right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-1-\sec \left (f x +e \right )\right )-2+2 \cos \left (f x +e \right )}{2 f \sqrt {\sin \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}}\) | \(214\) |
Input:
int(sin(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2/f/sin(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2)*((csc(f*x+e)-cot(f*x+e)+1)^(1 /2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*El lipticE((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))*(2+2*sec(f*x+e))+(csc (f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x +e)+cot(f*x+e))^(1/2)*EllipticF((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2 ))*(-1-sec(f*x+e))-2+2*cos(f*x+e))
\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {\sin \left (f x + e\right )}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b*sec(f*x + e))*sqrt(sin(f*x + e))/(b*sec(f*x + e)), x)
\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\sqrt {\sin {\left (e + f x \right )}}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \] Input:
integrate(sin(f*x+e)**(1/2)/(b*sec(f*x+e))**(1/2),x)
Output:
Integral(sqrt(sin(e + f*x))/sqrt(b*sec(e + f*x)), x)
\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {\sin \left (f x + e\right )}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(sin(f*x + e))/sqrt(b*sec(f*x + e)), x)
\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {\sin \left (f x + e\right )}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(sin(f*x + e))/sqrt(b*sec(f*x + e)), x)
Timed out. \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\sqrt {\sin \left (e+f\,x\right )}}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int(sin(e + f*x)^(1/2)/(b/cos(e + f*x))^(1/2),x)
Output:
int(sin(e + f*x)^(1/2)/(b/cos(e + f*x))^(1/2), x)
\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right )}d x \right )}{b} \] Input:
int(sin(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2),x)
Output:
(sqrt(b)*int((sqrt(sin(e + f*x))*sqrt(sec(e + f*x)))/sec(e + f*x),x))/b