\(\int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx\) [304]

Optimal result

Integrand size = 25, antiderivative size = 213 \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\frac {4}{a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}+\frac {4 \csc ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3963, 3957, 2954, 2952, 2647, 2720, 2644, 14, 2649} \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\frac {4 \csc ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}+\frac {4}{a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a^2 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 2952

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

Rule 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e +
f*x])^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 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))^2} \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x)}{(-a-a \cos (c+d x))^2} \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}\right ) \, dx}{a^4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\int \frac {\cos ^4(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {1-x^2}{x^{5/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 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^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {4 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \text {Subst}\left (\int \left (\frac {1}{x^{5/2}}-\frac {1}{\sqrt {x}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {4}{a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}+\frac {4 \csc ^2(c+d x)}{3 a^2 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 )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (12 (1+\cos (c+d x)) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )+(15+10 \cos (c+d x)-\cos (2 (c+d x))) \sqrt {\sin (c+d x)}\right )}{6 a^2 d (e \csc (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

(Sec[(c + d*x)/2]^2*(12*(1 + Cos[c + d*x])*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (15 + 10*Cos[c + d*x] - Cos[2
*(c + d*x)])*Sqrt[Sin[c + d*x]]))/(6*a^2*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 = 9.99 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.86

method	result	size
default	\(\frac {\sqrt {2}\, \left (6 i \sqrt {i \left (-i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \cos \left (d x +c \right )^{2}+12 i \sqrt {i \left (-i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+6 i \sqrt {i \left (-i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+\cos \left (d x +c \right )^{2} \sqrt {2}\, \sin \left (d x +c \right )-5 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}-8 \sqrt {2}\, \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 a^{2} d \left (\cos \left (d x +c \right )-1\right ) \left (\cos \left (d x +c \right )+1\right )^{2} e \sqrt {e \csc \left (d x +c \right )}}\)	\(397\)

[In]

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

[Out]

1/3/a^2/d*2^(1/2)*(6*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+12*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)+6*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*2^(1/2)*sin(d*x+c)-5*cos(d*x+c)*sin(d*x+c)*2^(1/2)-8*2^(1/2)*sin(d*x+c))/(cos(d*x+c)
-1)/(cos(d*x+c)+1)^2/e/(e*csc(d*x+c))^(1/2)*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 = 135, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left ({\left (\cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) - 8\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 3 \, \sqrt {2 i \, e} {\left (-i \, \cos \left (d x + c\right ) - i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {-2 i \, e} {\left (i \, \cos \left (d x + c\right ) + i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{3 \, {\left (a^{2} d e^{2} \cos \left (d x + c\right ) + a^{2} d e^{2}\right )}} \]

[In]

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

[Out]

-2/3*((cos(d*x + c)^2 - 5*cos(d*x + c) - 8)*sqrt(e/sin(d*x + c))*sin(d*x + c) + 3*sqrt(2*I*e)*(-I*cos(d*x + c)
- I)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + 3*sqrt(-2*I*e)*(I*cos(d*x + c) + I)*weierstra
ssPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)))/(a^2*d*e^2*cos(d*x + c) + a^2*d*e^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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