3.3.96 \(\int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx\) [296]

3.3.96.1 Optimal result

Integrand size = 25, antiderivative size = 99 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}-\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]

output

2*cot(d*x+c)/a/d/(e*csc(d*x+c))^(1/2)-2*csc(d*x+c)/a/d/(e*csc(d*x+c))^(1/2 
)-4*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*Elliptic 
E(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))/a/d/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^( 
1/2)
 

3.3.96.2 Mathematica [C] (verified)

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

Time = 1.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {6 (2 i+\cot (c+d x)-\csc (c+d x))-4 \sqrt {1-e^{2 i (c+d x)}} (i+\cot (c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )}{3 a d \sqrt {e \csc (c+d x)}} \]

input

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

output

(6*(2*I + Cot[c + d*x] - Csc[c + d*x]) - 4*Sqrt[1 - E^((2*I)*(c + d*x))]*( 
I + Cot[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])/( 
3*a*d*Sqrt[e*Csc[c + d*x]])
 

3.3.96.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 3318, 3042, 3044, 15, 3047, 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.

\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) \sqrt {e \csc (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \sqrt {e \sec \left (c+d x-\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4366

\(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x)}}{\sec (c+d x) a+a}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )}}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 4360

\(\displaystyle \frac {\int -\frac {\cos (c+d x) \sqrt {\sin (c+d x)}}{-\cos (c+d x) a-a}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {\cos (c+d x) \sqrt {\sin (c+d x)}}{\cos (c+d x) a+a}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\cos (c+d x) \sqrt {\sin (c+d x)}}{\cos (c+d x) a+a}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 3047

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

\(\displaystyle \frac {-\frac {2}{a d \sqrt {\sin (c+d x)}}-\frac {-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {2 \cos (c+d x)}{d \sqrt {\sin (c+d x)}}}{a}}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

input

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

output

(-(((-4*EllipticE[(c - Pi/2 + d*x)/2, 2])/d - (2*Cos[c + d*x])/(d*Sqrt[Sin 
[c + d*x]]))/a) - 2/(a*d*Sqrt[Sin[c + d*x]]))/(Sqrt[e*Csc[c + d*x]]*Sqrt[S 
in[c + d*x]])
 

3.3.96.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[1/(a*f)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a 
*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

rule 3047

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ 
(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1)))   Int[(a*Cos[e + f*x] 
)^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ 
[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
 

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]
 

rule 3318

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a   Int 
[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d)   Int 
[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, 
d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
 

rule 4360

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si 
n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
 

rule 4366

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( 
x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos 
[e + f*x]^FracPart[p]   Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / 
; FreeQ[{a, b, e, f, g, m, p}, x] &&  !IntegerQ[p]
 

3.3.96.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 7.46 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.36

method	result	size
default	\(-\frac {\sqrt {2}\, \left (4 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )-2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+4 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (d x +c \right )-\sqrt {2}\right ) \csc \left (d x +c \right )}{a d \sqrt {e \csc \left (d x +c \right )}}\)	\(432\)

input

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

output

-1/a/d*2^(1/2)*(4*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-c 
sc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE((-I*(I-cot( 
d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*cos(d*x+c)-2*(-I*(I-cot(d*x+c)+csc( 
d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d* 
x+c)))^(1/2)*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*c 
os(d*x+c)+4*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x 
+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE((-I*(I-cot(d*x+c) 
+csc(d*x+c)))^(1/2),1/2*2^(1/2))-2*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(- 
I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*Elli 
pticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))+2^(1/2)*cos(d*x+c) 
-2^(1/2))/(e*csc(d*x+c))^(1/2)*csc(d*x+c)
 

3.3.96.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {2 \, {\left (\sqrt {\frac {e}{\sin \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 1\right )} + \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{a d e} \]

input

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

output

2*(sqrt(e/sin(d*x + c))*(cos(d*x + c) - 1) + sqrt(2*I*e)*weierstrassZeta(4 
, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + sqrt(-2*I 
*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d 
*x + c))))/(a*d*e)
 

3.3.96.6 Sympy [F]

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

input

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

output

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

3.3.96.7 Maxima [F]

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

input

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

output

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

3.3.96.8 Giac [F]

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

input

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

output

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

3.3.96.9 Mupad [F(-1)]

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

input

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

output

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