Integrand size = 25, antiderivative size = 53 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \]
-(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x) ,2^(1/2))/b/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2)/sin(2*b*x+2*a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.81 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\frac {\sqrt [4]{-\cot ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\csc ^2(a+b x)\right ) \tan (a+b x)}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \]
((-Cot[a + b*x]^2)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, Csc[a + b*x]^2] *Tan[a + b*x])/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]])
Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3110, 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.
\(\displaystyle \int \frac {1}{\sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}dx\) |
\(\Big \downarrow \) 3110 |
\(\displaystyle \frac {\int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {\int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
EllipticE[a - Pi/4 + b*x, 2]/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]* Sqrt[Sin[2*a + 2*b*x]])
3.3.59.3.1 Defintions of rubi rules used
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[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n*(a*Sin[e + f*x] )^m*(b*Cos[e + f*x])^n Int[1/((a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[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. \(392\) vs. \(2(73)=146\).
Time = 1.31 (sec) , antiderivative size = 393, normalized size of antiderivative = 7.42
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (b x +a \right )^{2}-\sqrt {2}\, \cos \left (b x +a \right )\right ) \sec \left (b x +a \right ) \csc \left (b x +a \right )}{2 b \sqrt {c \sec \left (b x +a \right )}\, \sqrt {d \csc \left (b x +a \right )}}\) | \(393\) |
-1/2/b*2^(1/2)*(2*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a)+1 )^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^ (1/2),1/2*2^(1/2))*cos(b*x+a)-(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(cot(b*x+a)- csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)- cot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)+2*(1+csc(b*x+a)-cot(b*x+a))^(1/2 )*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE( (1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))-(1+csc(b*x+a)-cot(b*x+a))^(1/ 2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF ((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+2^(1/2)*cos(b*x+a)^2-2^(1/2) *cos(b*x+a))/(c*sec(b*x+a))^(1/2)/(d*csc(b*x+a))^(1/2)*sec(b*x+a)*csc(b*x+ a)
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int \frac {1}{\sqrt {c \sec {\left (a + b x \right )}} \sqrt {d \csc {\left (a + b x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}} \,d x \]