\(\int \frac {\csc ^2(e+f x)}{(a+b \sin ^2(e+f x))^{3/2}} \, dx\) [161]

Optimal result

Integrand size = 25, antiderivative size = 235 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {b \cot (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f}-\frac {(a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{a f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

b*cot(f*x+e)/a/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)-(a+2*b)*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^2/(a+b)/f-(a+2*b
)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*sin(f*x+e)^2)^(1/2)/a^2/(a+b)/f/(1+b
*sin(f*x+e)^2/a)^(1/2)+EllipticF(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(1+b*sin(f*x+e)^2/a)
^(1/2)/a/f/(a+b*sin(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 235, 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 = {3267, 483, 597, 538, 437, 435, 432, 430} \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{a^2 f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {(a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 f (a+b)}+\frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{a f \sqrt {a+b \sin ^2(e+f x)}}+\frac {b \cot (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]

[In]

Int[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

(b*Cot[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]
)/(a^2*(a + b)*f) - ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[
a + b*Sin[e + f*x]^2])/(a^2*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin
[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(a*f*Sqrt[a + b*Sin[e + f*x]^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 3267

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff^(m + 1)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[x^m*((a + b*ff^2*
x^2)^p/Sqrt[1 - ff^2*x^2]), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] &&  !In
tegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {b \cot (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a-2 b+b x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{a (a+b) f} \\ & = \frac {b \cot (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a b-b (a+2 b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{a^2 (a+b) f} \\ & = \frac {b \cot (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{a f}+\frac {\left ((-a-2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{a^2 (a+b) f} \\ & = \frac {b \cot (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f}+\frac {\left ((-a-2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{a^2 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{a f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {b \cot (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f}-\frac {(a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a^2 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{a f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.72 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\left (-2 a^2-3 a b-2 b^2+b (a+2 b) \cos (2 (e+f x))\right ) \cot (e+f x)-\sqrt {2} a (a+2 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+\sqrt {2} a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{\sqrt {2} a^2 (a+b) f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]

[In]

Integrate[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

((-2*a^2 - 3*a*b - 2*b^2 + b*(a + 2*b)*Cos[2*(e + f*x)])*Cot[e + f*x] - Sqrt[2]*a*(a + 2*b)*Sqrt[(2*a + b - b*
Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*Ell
ipticF[e + f*x, -(b/a)])/(Sqrt[2]*a^2*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\left (a b +2 b^{2}\right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-a^{2}-2 a b -2 b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a \left (F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right )}{a^{2} \sin \left (f x +e \right ) \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\)	\(199\)

[In]

int(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((a*b+2*b^2)*cos(f*x+e)^4+(-a^2-2*a*b-2*b^2)*cos(f*x+e)^2+sin(f*x+e)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*
x+e)^2)^(1/2)*a*(EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a+EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-EllipticE(sin(f
*x+e),(-1/a*b)^(1/2))*a-2*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*b))/a^2/sin(f*x+e)/(a+b)/cos(f*x+e)/(a+b*sin(f*
x+e)^2)^(1/2)/f

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 1075, normalized size of antiderivative = 4.57 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*((2*(I*a^2*b + 3*I*a*b^2 + 2*I*b^3 + (-I*a*b^2 - 2*I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*s
in(f*x + e) - (-2*I*a^3 - 7*I*a^2*b - 7*I*a*b^2 - 2*I*b^3 + (2*I*a^2*b + 5*I*a*b^2 + 2*I*b^3)*cos(f*x + e)^2)*
sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a
*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*
b)/b^2))/b^2) + (2*(-I*a^2*b - 3*I*a*b^2 - 2*I*b^3 + (I*a*b^2 + 2*I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 +
a*b)/b^2)*sin(f*x + e) - (2*I*a^3 + 7*I*a^2*b + 7*I*a*b^2 + 2*I*b^3 + (-2*I*a^2*b - 5*I*a*b^2 - 2*I*b^3)*cos(f
*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sq
rt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqr
t((a^2 + a*b)/b^2))/b^2) - 2*(4*(I*a^2*b + 2*I*a*b^2 + I*b^3 + (-I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqr
t((a^2 + a*b)/b^2)*sin(f*x + e) + (-2*I*a^3 - 3*I*a^2*b - I*a*b^2 + (2*I*a^2*b + I*a*b^2)*cos(f*x + e)^2)*sqrt
(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/
b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b
^2))/b^2) - 2*(4*(-I*a^2*b - 2*I*a*b^2 - I*b^3 + (I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b
^2)*sin(f*x + e) + (2*I*a^3 + 3*I*a^2*b + I*a*b^2 + (-2*I*a^2*b - I*a*b^2)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x +
e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)
/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 2*(
(a*b^2 + 2*b^3)*cos(f*x + e)^3 - (a^2*b + 2*a*b^2 + 2*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(((a
^3*b^2 + a^2*b^3)*f*cos(f*x + e)^2 - (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)*sin(f*x + e))

Sympy [F]

\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(csc(f*x+e)**2/(a+b*sin(f*x+e)**2)**(3/2),x)

[Out]

Integral(csc(e + f*x)**2/(a + b*sin(e + f*x)**2)**(3/2), x)

Maxima [F]

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

[In]

integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)

Giac [F]

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

[In]

integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(1/(sin(e + f*x)^2*(a + b*sin(e + f*x)^2)^(3/2)),x)

[Out]

int(1/(sin(e + f*x)^2*(a + b*sin(e + f*x)^2)^(3/2)), x)