\(\int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx\) [297]

Optimal result

Integrand size = 25, antiderivative size = 106 \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\frac {2}{a d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x)}{3 a d e \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 a d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]

[Out]

2/a/d/e/(e*csc(d*x+c))^(1/2)-2/3*cos(d*x+c)/a/d/e/(e*csc(d*x+c))^(1/2)+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))/a/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^
(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 106, 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, 2649, 2720} \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\frac {2}{a d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x)}{3 a d e \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 a d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]

[In]

Int[1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])),x]

[Out]

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

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 2649

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

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )-2 (-3+\cos (c+d x)) \sqrt {\sin (c+d x)}}{3 a d (e \csc (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)} \]

[In]

Integrate[1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 10.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.59

method	result	size
default	\(\frac {\sqrt {2}\, \left (2 i \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \cos \left (d x +c \right )+2 i \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}+\cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}-3 \sqrt {2}\, \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 a d \sqrt {e \csc \left (d x +c \right )}\, e \left (\cos \left (d x +c \right )-1\right ) \left (\cos \left (d x +c \right )+1\right )}\)	\(275\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=-\frac {2 \, {\left (\sqrt {\frac {e}{\sin \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 3\right )} \sin \left (d x + c\right ) - i \, \sqrt {2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {-2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{3 \, a d e^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\frac {\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )} + \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a} \]

[In]

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

[Out]

Integral(1/((e*csc(c + d*x))**(3/2)*sec(c + d*x) + (e*csc(c + d*x))**(3/2)), x)/a

Maxima [F]

\[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

[In]

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

[Out]

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