Integrand size = 25, antiderivative size = 234 \[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {(2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {(2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {2 (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}} \]
-1/3*(2*a+b)*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a/f-1/3*cot(f*x+e)*csc(f* x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)/f-1/3*(2*a+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/f/(1+b*s in(f*x+e)^2/a)^(1/2)+2/3*(a+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)/f/(a+b*sin(f*x+e)^2)^(1 /2)
Time = 2.21 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.80 \[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\frac {\left (4 \left (2 a^2+4 a b+b^2\right ) \cos (2 (e+f x))-(2 a+b) (8 a+3 b+b \cos (4 (e+f x)))\right ) \cot (e+f x) \csc ^2(e+f x)}{2 \sqrt {2}}-2 a (2 a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+4 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{6 a f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(((4*(2*a^2 + 4*a*b + b^2)*Cos[2*(e + f*x)] - (2*a + b)*(8*a + 3*b + b*Cos [4*(e + f*x)]))*Cot[e + f*x]*Csc[e + f*x]^2)/(2*Sqrt[2]) - 2*a*(2*a + b)*S qrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 4*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6 *a*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])
Time = 0.45 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3667, 377, 445, 25, 27, 399, 323, 321, 330, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)^2}}{\sin (e+f x)^4}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\csc ^4(e+f x) \sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \int \frac {\csc ^2(e+f x) \left (b \sin ^2(e+f x)+2 a+b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (-\frac {\int -\frac {b \left (a-(2 a+b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {\int \frac {b \left (a-(2 a+b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {b \int \frac {a-(2 a+b) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {b \left (\frac {2 a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}-\frac {(2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}\right )}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {b \left (\frac {2 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}\right )}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {b \left (\frac {2 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}\right )}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {b \left (\frac {2 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}\right )}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {b \left (\frac {2 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}\right )}{a}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )-\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \csc ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(-1/3*(Csc[e + f*x]^3*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2]) + (-(((2*a + b)*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/a) + (b*(-(((2*a + b)*Elliptic E[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*Sqrt[1 + (b *Sin[e + f*x]^2)/a])) + (2*a*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a )]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*Sqrt[a + b*Sin[e + f*x]^2])))/a)/3)) /f
3.2.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(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] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[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]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 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]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[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] && !IntegerQ[p]
Time = 2.46 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )+2 b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \left (\sin ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \left (\sin ^{3}\left (f x +e \right )\right )+2 a b \left (\sin ^{6}\left (f x +e \right )\right )+b^{2} \left (\sin ^{6}\left (f x +e \right )\right )+2 a^{2} \left (\sin ^{4}\left (f x +e \right )\right )-b^{2} \left (\sin ^{4}\left (f x +e \right )\right )-a^{2} \left (\sin ^{2}\left (f x +e \right )\right )-2 a b \left (\sin ^{2}\left (f x +e \right )\right )-a^{2}}{3 a \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(342\) |
1/3*(2*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x +e),(-1/a*b)^(1/2))*a^2*sin(f*x+e)^3+2*b*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f* x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a*sin(f*x+e)^3-2*(co s(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a* b)^(1/2))*a^2*sin(f*x+e)^3-(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/ 2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b*sin(f*x+e)^3+2*a*b*sin(f*x+e)^ 6+b^2*sin(f*x+e)^6+2*a^2*sin(f*x+e)^4-b^2*sin(f*x+e)^4-a^2*sin(f*x+e)^2-2* a*b*sin(f*x+e)^2-a^2)/a/sin(f*x+e)^3/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 947, normalized size of antiderivative = 4.05 \[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\text {Too large to display} \]
1/6*((2*((-2*I*a*b - I*b^2)*cos(f*x + e)^2 + 2*I*a*b + I*b^2)*sqrt(-b)*sqr t((a^2 + a*b)/b^2)*sin(f*x + e) - ((4*I*a^2 + 4*I*a*b + I*b^2)*cos(f*x + e )^2 - 4*I*a^2 - 4*I*a*b - I*b^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*((2*I*a*b + I*b^2)*cos(f *x + e)^2 - 2*I*a*b - I*b^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - ((-4*I*a^2 - 4*I*a*b - I*b^2)*cos(f*x + e)^2 + 4*I*a^2 + 4*I*a*b + I*b^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*(2*((-I*a*b - I*b^2)*cos(f*x + e)^2 + I*a*b + I*b^2)*sqrt (-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + ((-2*I*a^2 - I*a*b)*cos(f*x + e) ^2 + 2*I*a^2 + I*a*b)*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*(2*((I*a*b + I*b^2)*cos(f*x + e)^2 - I*a*b - I*b^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + ((2*I*a^2 + I*a*b)*cos(f*x + e)^2 - 2*I*a^2 - I*a*b)*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...
\[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \csc ^{4}{\left (e + f x \right )}\, dx \]
\[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \csc \left (f x + e\right )^{4} \,d x } \]
\[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \csc \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \frac {\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}}{{\sin \left (e+f\,x\right )}^4} \,d x \]