\(\int \frac {\csc ^2(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [170]

Optimal result

Integrand size = 25, antiderivative size = 322 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(3 a+4 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

1/3*b*cot(f*x+e)/a/(a+b)/f/(a+b*sin(f*x+e)^2)^(3/2)+2/3*b*(3*a+2*b)*cot(f*x+e)/a^2/(a+b)^2/f/(a+b*sin(f*x+e)^2
)^(1/2)-1/3*(3*a^2+13*a*b+8*b^2)*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^3/(a+b)^2/f-1/3*(3*a^2+13*a*b+8*b^2)*El
lipticE(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^3/(a+b)^2/f/(1+b*s
in(f*x+e)^2/a)^(1/2)+1/3*(3*a+4*b)*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^2/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3267, 483, 593, 597, 538, 437, 435, 432, 430} \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(3 a+4 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 a^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^3 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)^2}+\frac {b \cot (e+f x)}{3 a f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]

[In]

Int[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]

[Out]

(b*Cot[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(3*a + 2*b)*Cot[e + f*x])/(3*a^2*(a + b)^
2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a^2 + 13*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a
+ b)^2*f) - ((3*a^2 + 13*a*b + 8*b^2)*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])/(3*a^3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((3*a + 4*b)*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])/(3*a^2*(a + b)*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 593

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

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 )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 a-4 b+3 b x^2}{x^2 \sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a (a+b) f} \\ & = \frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 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 {3 a^2+13 a b+8 b^2-2 b (3 a+2 b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b)^2 f} \\ & = \frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {2 a b (3 a+2 b)+b \left (3 a^2+13 a b+8 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b)^2 f} \\ & = \frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}+\frac {\left ((3 a+4 b) \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 )}{3 a^2 (a+b) f}-\frac {\left (\left (3 a^2+13 a b+8 b^2\right ) \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 )}{3 a^3 (a+b)^2 f} \\ & = \frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (\left (3 a^2+13 a b+8 b^2\right ) \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 )}{3 a^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left ((3 a+4 b) \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 )}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \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)}}{3 a^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(3 a+4 b) \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}}}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.77 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.66 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {4 a^2 \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \left (-\left (\left (3 a^2+13 a b+8 b^2\right ) E\left (e+f x\left |-\frac {b}{a}\right .\right )\right )+\left (3 a^2+7 a b+4 b^2\right ) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )-2 \sqrt {2} \left (3 (a+b)^2 (2 a+b-b \cos (2 (e+f x)))^2 \cot (e+f x)+2 a b^2 (a+b) \sin (2 (e+f x))+b^2 (7 a+5 b) (2 a+b-b \cos (2 (e+f x))) \sin (2 (e+f x))\right )}{12 a^3 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]

[In]

Integrate[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]

[Out]

(4*a^2*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(-((3*a^2 + 13*a*b + 8*b^2)*EllipticE[e + f*x, -(b/a)]) + (3*a
^2 + 7*a*b + 4*b^2)*EllipticF[e + f*x, -(b/a)]) - 2*Sqrt[2]*(3*(a + b)^2*(2*a + b - b*Cos[2*(e + f*x)])^2*Cot[
e + f*x] + 2*a*b^2*(a + b)*Sin[2*(e + f*x)] + b^2*(7*a + 5*b)*(2*a + b - b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])
)/(12*a^3*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))

Maple [A] (verified)

Time = 3.70 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {\left (-3 a^{2} b^{2}-13 a \,b^{3}-8 b^{4}\right ) \left (\cos ^{6}\left (f x +e \right )\right )+\left (6 a^{3} b +26 a^{2} b^{2}+38 a \,b^{3}+16 b^{4}\right ) \left (\cos ^{4}\left (f x +e \right )\right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a b \left (3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+7 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-3 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-13 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-3 a^{4}-12 a^{3} b -26 a^{2} b^{2}-25 a \,b^{3}-8 b^{4}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a \left (3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+10 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +11 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-3 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-16 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -21 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}\right ) \sin \left (f x +e \right )}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a +b \right )^{2} \sin \left (f x +e \right ) a^{3} \cos \left (f x +e \right ) f}\)	\(527\)

[In]

int(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*((-3*a^2*b^2-13*a*b^3-8*b^4)*cos(f*x+e)^6+(6*a^3*b+26*a^2*b^2+38*a*b^3+16*b^4)*cos(f*x+e)^4-(cos(f*x+e)^2)
^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*a*b*(3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2+7*EllipticF(sin(f*x+e
),(-1/a*b)^(1/2))*a*b+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-3*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2-13
*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b-8*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*b^2)*cos(f*x+e)^2*sin(f*x+e)+
(-3*a^4-12*a^3*b-26*a^2*b^2-25*a*b^3-8*b^4)*cos(f*x+e)^2+(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2
)*a*(3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^3+10*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b+11*EllipticF(sin
(f*x+e),(-1/a*b)^(1/2))*a*b^2+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^3-3*EllipticE(sin(f*x+e),(-1/a*b)^(1/2)
)*a^3-16*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b-21*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2-8*EllipticE(
sin(f*x+e),(-1/a*b)^(1/2))*b^3)*sin(f*x+e))/(a+b*sin(f*x+e)^2)^(3/2)/(a+b)^2/sin(f*x+e)/a^3/cos(f*x+e)/f

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 1719, normalized size of antiderivative = 5.34 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

1/6*((2*(-3*I*a^4*b - 19*I*a^3*b^2 - 37*I*a^2*b^3 - 29*I*a*b^4 - 8*I*b^5 + (-3*I*a^2*b^3 - 13*I*a*b^4 - 8*I*b^
5)*cos(f*x + e)^4 - 2*(-3*I*a^3*b^2 - 16*I*a^2*b^3 - 21*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2
+ a*b)/b^2)*sin(f*x + e) - (6*I*a^5 + 41*I*a^4*b + 93*I*a^3*b^2 + 95*I*a^2*b^3 + 45*I*a*b^4 + 8*I*b^5 + (6*I*a
^3*b^2 + 29*I*a^2*b^3 + 29*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^4 + 2*(-6*I*a^4*b - 35*I*a^3*b^2 - 58*I*a^2*b^3 - 3
7*I*a*b^4 - 8*I*b^5)*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)*elli
ptic_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*(3*I*a^4*b + 19*I*a^3*b^2 + 37*I*a^2*b^3 + 29*I*a*b^4
+ 8*I*b^5 + (3*I*a^2*b^3 + 13*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^4 - 2*(3*I*a^3*b^2 + 16*I*a^2*b^3 + 21*I*a*b^4 +
8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (-6*I*a^5 - 41*I*a^4*b - 93*I*a^3*b^2
- 95*I*a^2*b^3 - 45*I*a*b^4 - 8*I*b^5 + (-6*I*a^3*b^2 - 29*I*a^2*b^3 - 29*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 +
2*(6*I*a^4*b + 35*I*a^3*b^2 + 58*I*a^2*b^3 + 37*I*a*b^4 + 8*I*b^5)*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) - 4*((-9*I*a^
4*b - 35*I*a^3*b^2 - 51*I*a^2*b^3 - 33*I*a*b^4 - 8*I*b^5 + (-9*I*a^2*b^3 - 17*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^
4 + 2*(9*I*a^3*b^2 + 26*I*a^2*b^3 + 25*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f
*x + e) + (6*I*a^5 + 19*I*a^4*b + 22*I*a^3*b^2 + 11*I*a^2*b^3 + 2*I*a*b^4 + (6*I*a^3*b^2 + 7*I*a^2*b^3 + 2*I*a
*b^4)*cos(f*x + e)^4 + 2*(-6*I*a^4*b - 13*I*a^3*b^2 - 9*I*a^2*b^3 - 2*I*a*b^4)*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) -
4*((9*I*a^4*b + 35*I*a^3*b^2 + 51*I*a^2*b^3 + 33*I*a*b^4 + 8*I*b^5 + (9*I*a^2*b^3 + 17*I*a*b^4 + 8*I*b^5)*cos
(f*x + e)^4 + 2*(-9*I*a^3*b^2 - 26*I*a^2*b^3 - 25*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)
/b^2)*sin(f*x + e) + (-6*I*a^5 - 19*I*a^4*b - 22*I*a^3*b^2 - 11*I*a^2*b^3 - 2*I*a*b^4 + (-6*I*a^3*b^2 - 7*I*a^
2*b^3 - 2*I*a*b^4)*cos(f*x + e)^4 + 2*(6*I*a^4*b + 13*I*a^3*b^2 + 9*I*a^2*b^3 + 2*I*a*b^4)*cos(f*x + e)^2)*sqr
t(-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*((3*a^2*b^3 + 13*a*b^4 + 8*b^5)*cos(f*x + e)^5 - 2*(3*a^3*b^2 + 13*a^2*b^3 + 19*a*b^4 + 8*b^5)*
cos(f*x + e)^3 + (3*a^4*b + 12*a^3*b^2 + 26*a^2*b^3 + 25*a*b^4 + 8*b^5)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 +
a + b))/(((a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*f*cos(f*x + e)^4 - 2*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f*
cos(f*x + e)^2 + (a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*f)*sin(f*x + e))

Sympy [F]

\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(csc(f*x+e)**2/(a+b*sin(f*x+e)**2)**(5/2),x)

[Out]

Integral(csc(e + f*x)**2/(a + b*sin(e + f*x)**2)**(5/2), x)

Maxima [F]

\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)

Giac [F]

\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]

[In]

int(1/(sin(e + f*x)^2*(a + b*sin(e + f*x)^2)^(5/2)),x)

[Out]

int(1/(sin(e + f*x)^2*(a + b*sin(e + f*x)^2)^(5/2)), x)