\(\int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx\) [295]

Optimal result

Integrand size = 25, antiderivative size = 105 \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}-\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a d} \]

[Out]

2/3*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/a/d-2/3*csc(d*x+c)*(e*csc(d*x+c))^(1/2)/a/d-4/3*(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)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+
c)^(1/2)/a/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3963, 3957, 2918, 2644, 30, 2647, 2720} \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {e \csc (c+d x)}}{3 a d} \]

[In]

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

[Out]

(2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a*d) - (2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a*d) + (4*Sqrt[e*Csc[
c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a*d)

Rule 30

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

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[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] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2647

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] + Dist[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] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2720

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]

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[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 3957

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

Rule 3963

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_), x_Symbol] :> Dist[g^Int
Part[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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x)) \sqrt {\sin (c+d x)}} \, dx \\ & = -\left (\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) \sqrt {\sin (c+d x)}} \, dx\right ) \\ & = \frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a}-\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a} \\ & = \frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a}+\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\sin (c+d x)\right )}{a d} \\ & = \frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}-\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {2 (e \csc (c+d x))^{3/2} \left (-1+\cos (c+d x)-2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{3 a d e} \]

[In]

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

[Out]

(2*(e*Csc[c + d*x])^(3/2)*(-1 + Cos[c + d*x] - 2*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(3/2)))/(3*a
*d*e)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.08 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.71

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}}\, \left (1-\cos \left (d x +c \right )\right ) \left (2 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \csc \left (d x +c \right )}{3 a d \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\)	\(285\)

[In]

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

[Out]

1/3/a/d*2^(1/2)*(e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)+sin(d*x+c)))^(1/2)*(1-cos(d*x+c))*(2*I*(-I*(I-c
ot(d*x+c)+csc(d*x+c)))^(1/2)*2^(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)*E
llipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-(1-cos(d*x+c))^3*csc(d*x+c)^3-csc(d*x+c)+cot(d*x+c)
)/((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)*csc(d*x+c))^(1/2)/((1-cos(d*x+c))^3*csc(d*x+c)^3+csc(d*x+c
)-cot(d*x+c))^(1/2)*csc(d*x+c)

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {2 i \, e} {\left (i \, \cos \left (d x + c\right ) + i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {-2 i \, e} {\left (-i \, \cos \left (d x + c\right ) - i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{3 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]

[In]

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

[Out]

-2/3*(sqrt(2*I*e)*(I*cos(d*x + c) + I)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + sqrt(-2*I*e)
*(-I*cos(d*x + c) - I)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + sqrt(e/sin(d*x + c))*sin(d*x
+ c))/(a*d*cos(d*x + c) + a*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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