\(\int \frac {\sqrt {c \sec (a+b x)}}{(d \csc (a+b x))^{3/2}} \, dx\) [236]

Optimal result

Integrand size = 25, antiderivative size = 93 \[ \int \frac {\sqrt {c \sec (a+b x)}}{(d \csc (a+b x))^{3/2}} \, dx=-\frac {c}{b d \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {d \csc (a+b x)} \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{2 b d^2} \] Output:

-c/b/d/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2)+1/2*(d*csc(b*x+a))^(1/2)* 
InverseJacobiAM(a-1/4*Pi+b*x,2^(1/2))*(c*sec(b*x+a))^(1/2)*sin(2*b*x+2*a)^ 
(1/2)/b/d^2
 

Mathematica [C] (verified)

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

Time = 0.88 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c \sec (a+b x)}}{(d \csc (a+b x))^{3/2}} \, dx=-\frac {\left (1+\cos (2 (a+b x))+\left (-\cot ^2(a+b x)\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\csc ^2(a+b x)\right )\right ) (c \sec (a+b x))^{3/2}}{2 b c d \sqrt {d \csc (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 93, 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, 3107, 3042, 3110, 3042, 3053, 3042, 3120}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3107

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3110

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3053

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{2 b d^2}-\frac {c}{b d \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\)

Input:

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

Output:

-(c/(b*d*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]])) + (Sqrt[d*Csc[a + b*x 
]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x] 
])/(2*b*d^2)
 

Defintions of rubi rules used

rule 3042

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

rule 3053

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ 
)]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b 
*Cos[e + f*x]])   Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f 
}, x]
 

rule 3107

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n 
_.), x_Symbol] :> Simp[b*(a*Csc[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1) 
/(a*f*(m + n))), x] + Simp[(m + 1)/(a^2*(m + n))   Int[(a*Csc[e + f*x])^(m 
+ 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] 
&& NeQ[m + n, 0] && 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 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.45

method	result	size
default	\(\frac {\left (-\frac {\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {2 \cot \left (b x +a \right )-2 \csc \left (b x +a \right )+2}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (b x +a \right )-\csc \left (b x +a \right )\right )}{2}-\cos \left (b x +a \right )\right ) \sqrt {c \sec \left (b x +a \right )}}{b d \sqrt {d \csc \left (b x +a \right )}}\)	\(135\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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