\(\int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx\) [243]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \] Output:

2*c*d*(c*sec(b*x+a))^(1/2)/b/(d*csc(b*x+a))^(3/2)+2*c^2*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 = 0.55 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=-\frac {2 c d \left (-1+\sqrt [4]{-\cot ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\csc ^2(a+b x)\right )\right ) \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}} \] Input:

Integrate[(c*Sec[a + b*x])^(3/2)/Sqrt[d*Csc[a + b*x]],x]
 

Output:

(-2*c*d*(-1 + (-Cot[a + b*x]^2)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, Cs 
c[a + b*x]^2])*Sqrt[c*Sec[a + b*x]])/(b*(d*Csc[a + b*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3106, 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.

\(\displaystyle \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}}dx\)

\(\Big \downarrow \) 3106

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-2 c^2 \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-2 c^2 \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}dx\)

\(\Big \downarrow \) 3110

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}\)

\(\Big \downarrow \) 3052

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\)

Input:

Int[(c*Sec[a + b*x])^(3/2)/Sqrt[d*Csc[a + b*x]],x]
 

Output:

(2*c*d*Sqrt[c*Sec[a + b*x]])/(b*(d*Csc[a + b*x])^(3/2)) - (2*c^2*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 3106

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[a*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 
1)/(f*(n - 1))), x] + Simp[b^2*((m + n - 2)/(n - 1))   Int[(a*Csc[e + f*x]) 
^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 
1] && IntegersQ[2*m, 2*n]
 

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. \(221\) vs. \(2(80)=160\).

Time = 1.61 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.49

method	result	size
default	\(\frac {\left (2-2 \cos \left (b x +a \right )+\sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {2 \cot \left (b x +a \right )-2 \csc \left (b x +a \right )+2}\, \left (-\cos \left (b x +a \right )-1\right ) \operatorname {EllipticF}\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (2 \cos \left (b x +a \right )+2\right ) \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {2 \cot \left (b x +a \right )-2 \csc \left (b x +a \right )+2}\, \operatorname {EllipticE}\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {c \sec \left (b x +a \right )}\, c \csc \left (b x +a \right )}{b \sqrt {d \csc \left (b x +a \right )}}\)	\(222\)

Input:

int((c*sec(b*x+a))^(3/2)/(d*csc(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/b*(2-2*cos(b*x+a)+(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)*(-cos(b*x+a)-1)*EllipticF((-c 
ot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))+(2*cos(b*x+a)+2)*(cot(b*x+a)-cs 
c(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)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2)))*(c*sec 
(b*x+a))^(1/2)*c/(d*csc(b*x+a))^(1/2)*csc(b*x+a)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.74 \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\frac {\sqrt {-4 i \, c d} c E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + \sqrt {4 i \, c d} c E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - \sqrt {-4 i \, c d} c F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) - \sqrt {4 i \, c d} c F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - 4 \, {\left (c \cos \left (b x + a\right )^{2} - c\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{2 \, b d} \] Input:

integrate((c*sec(b*x+a))^(3/2)/(d*csc(b*x+a))^(1/2),x, algorithm="fricas")
 

Output:

1/2*(sqrt(-4*I*c*d)*c*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1 
) + sqrt(4*I*c*d)*c*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) 
- sqrt(-4*I*c*d)*c*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) - 
 sqrt(4*I*c*d)*c*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - 4 
*(c*cos(b*x + a)^2 - c)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)))/(b*d)
 

Sympy [F]

\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int \frac {\left (c \sec {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\sqrt {d \csc {\left (a + b x \right )}}}\, dx \] Input:

integrate((c*sec(b*x+a))**(3/2)/(d*csc(b*x+a))**(1/2),x)
 

Output:

Integral((c*sec(a + b*x))**(3/2)/sqrt(d*csc(a + b*x)), x)
 

Maxima [F]

\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int { \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \csc \left (b x + a\right )}} \,d x } \] Input:

integrate((c*sec(b*x+a))^(3/2)/(d*csc(b*x+a))^(1/2),x, algorithm="maxima")
 

Output:

integrate((c*sec(b*x + a))^(3/2)/sqrt(d*csc(b*x + a)), x)
 

Giac [F]

\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int { \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \csc \left (b x + a\right )}} \,d x } \] Input:

integrate((c*sec(b*x+a))^(3/2)/(d*csc(b*x+a))^(1/2),x, algorithm="giac")
 

Output:

integrate((c*sec(b*x + a))^(3/2)/sqrt(d*csc(b*x + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int \frac {{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}}{\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}} \,d x \] Input:

int((c/cos(a + b*x))^(3/2)/(d/sin(a + b*x))^(1/2),x)
 

Output:

int((c/cos(a + b*x))^(3/2)/(d/sin(a + b*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}\, \sec \left (b x +a \right )}{\csc \left (b x +a \right )}d x \right ) c}{d} \] Input:

int((c*sec(b*x+a))^(3/2)/(d*csc(b*x+a))^(1/2),x)
 

Output:

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