\(\int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx\) [259]

Optimal result

Integrand size = 25, antiderivative size = 53 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \] Output:

-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)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\frac {\sqrt [4]{-\cot ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\csc ^2(a+b x)\right ) \tan (a+b x)}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \] Input:

Integrate[1/(Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]),x]
 

Output:

((-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]])
 

Rubi [A] (verified)

Time = 0.37 (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

\(\displaystyle \frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\)

Input:

Int[1/(Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]),x]
 

Output:

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]])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3052

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]
 

rule 3110

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]
 

rule 3119

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(49)=98\).

Time = 1.37 (sec) , antiderivative size = 236, normalized size of antiderivative = 4.45

method	result	size
default	\(-\frac {\left (\left (2 \cos \left (b x +a \right )+2\right ) \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {2 \cot \left (b x +a \right )-2 \csc \left (b x +a \right )+2}\, \operatorname {EllipticE}\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (-\cos \left (b x +a \right )-1\right ) \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {2 \cot \left (b x +a \right )-2 \csc \left (b x +a \right )+2}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\cos \left (b x +a \right ) \left (-2+2 \cos \left (b x +a \right )\right )\right ) \sec \left (b x +a \right ) \csc \left (b x +a \right )}{2 b \sqrt {c \sec \left (b x +a \right )}\, \sqrt {d \csc \left (b x +a \right )}}\)	\(236\)

Input:

int(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2/b*((2*cos(b*x+a)+2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-cot(b*x+a)+csc(b* 
x+a)+1)^(1/2)*(2*cot(b*x+a)-2*csc(b*x+a)+2)^(1/2)*EllipticE((-cot(b*x+a)+c 
sc(b*x+a)+1)^(1/2),1/2*2^(1/2))+(-cos(b*x+a)-1)*(-cot(b*x+a)+csc(b*x+a)+1) 
^(1/2)*(2*cot(b*x+a)-2*csc(b*x+a)+2)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*E 
llipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))+cos(b*x+a)*(-2+2*co 
s(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)
 

Fricas [F]

\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2),x, algorithm="fricas 
")
 

Output:

integral(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))/(c*d*csc(b*x + a)*sec(b 
*x + a)), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int \frac {1}{\sqrt {c \sec {\left (a + b x \right )}} \sqrt {d \csc {\left (a + b x \right )}}}\, dx \] Input:

integrate(1/(d*csc(b*x+a))**(1/2)/(c*sec(b*x+a))**(1/2),x)
 

Output:

Integral(1/(sqrt(c*sec(a + b*x))*sqrt(d*csc(a + b*x))), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(1/(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}} \,d x \] Input:

int(1/((c/cos(a + b*x))^(1/2)*(d/sin(a + b*x))^(1/2)),x)
 

Output:

int(1/((c/cos(a + b*x))^(1/2)*(d/sin(a + b*x))^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}}{\csc \left (b x +a \right ) \sec \left (b x +a \right )}d x \right )}{c d} \] Input:

int(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2),x)
 

Output:

(sqrt(d)*sqrt(c)*int((sqrt(sec(a + b*x))*sqrt(csc(a + b*x)))/(csc(a + b*x) 
*sec(a + b*x)),x))/(c*d)