\(\int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx\) [245]

Optimal result

Integrand size = 25, antiderivative size = 776 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \]

[Out]

-1/4*arctan(cos(d*x+c)*(-a)^(1/2)/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2))/d/(-a)^(1/2)-1/2*cot(d*x+c)*csc
(d*x+c)*(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2)/a/d-1/2*cos(d*x+c)*b^(1/2)*(a+b-2*b*cos(d*x+c)^2+b*cos(d*x
+c)^4)^(1/2)/a/d/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))/(a+b)^(1/2)+1/2*b^(1/4)*(a+b)^(3/4)*(cos(2*arctan(b^(1/4
)*cos(d*x+c)/(a+b)^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))*EllipticE(sin(2*arctan(b^(1/
4)*cos(d*x+c)/(a+b)^(1/4))),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))*(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))*((a+b-2*
b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))^2)^(1/2)/a/d/(a+b-2*b*cos(d*x+c)^2+b
*cos(d*x+c)^4)^(1/2)-1/8*(a+b)^(1/4)*(cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1
/4)*cos(d*x+c)/(a+b)^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1/4*(b^(1/2)+(a+b)^(1/2)
)^2/b^(1/2)/(a+b)^(1/2),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))*(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))*(b^(1/2)-(a+
b)^(1/2))^2*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))^2)^(1/2)/a/b^(1/
4)/d/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2)-1/2*b^(1/4)*(cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))^2)
^(1/2)/cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1
/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))*(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))*(a+b-b^(1/2)*(a+b)^(1/2))*((a+b-2*b*c
os(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))^2)^(1/2)/a/(a+b)^(1/4)/d/(a+b-2*b*cos(d
*x+c)^2+b*cos(d*x+c)^4)^(1/2)

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 776, 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 = {3294, 1237, 1728, 1209, 1722, 1117, 1720} \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt [4]{b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{2 a d \sqrt [4]{a+b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{2 a d \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d \sqrt {a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d} \]

[In]

Int[Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]

[Out]

-1/4*ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]]/(Sqrt[-a]*d) - (Sqrt[
b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(2*a*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c +
d*x]^2)/Sqrt[a + b])) - (Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]*Cot[c + d*x]*Csc[c + d*x])/(2*a*d
) + (b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos
[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])
/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*a*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - (
b^(1/4)*(a + b - Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x
]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Co
s[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*a*(a + b)^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2
+ b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sq
rt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*Ell
ipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1
+ Sqrt[b]/Sqrt[a + b])/2])/(8*a*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1237

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> Simp[(-e^2)*x*(d + e*x^2
)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b
*d*e + a*e^2)), Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q +
1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ
[b^2 - 4*a*c, 0] && ILtQ[q, -1]

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1722

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With
[{q = Rt[c/a, 2]}, Dist[(A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x],
x] + Dist[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2)), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]
, x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2
- a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]

Rule 1728

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]
, A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/(e*q), Int[(1 - q*x^2)/Sqrt[a + b
*x^2 + c*x^4], x], x] + Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[
a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[
c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] &&  !GtQ[b^2 - 4*a*c, 0]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\text {Subst}\left (\int \frac {-a+b-2 b x^2+b x^4}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\text {Subst}\left (\int \frac {-b (-a+b)+b^{3/2} \sqrt {a+b}+\left (2 b^2-b \left (b+\sqrt {b} \sqrt {a+b}\right )\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a b d}+\frac {\left (\sqrt {b} \sqrt {a+b}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}+\frac {\left (\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right )\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {\left (\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{a \sqrt {a+b} d} \\ & = -\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 25.16 (sec) , antiderivative size = 1442, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\sqrt {8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))} \cot (c+d x) \csc (c+d x)}{4 \sqrt {2} a d}+\frac {\sec (c+d x) \left (-a-2 a \tan ^2(c+d x)-a \tan ^4(c+d x)-b \tan ^4(c+d x)-\frac {b \left (\sqrt {b} E\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )+i \left (\sqrt {a}+i \sqrt {b}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}} \left (\sqrt {b} \tan ^2(c+d x)+i \sqrt {a} \left (1+\tan ^2(c+d x)\right )\right )}{(a+b) \sqrt {-\frac {i \left (a+i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}+\frac {a \left (-i \sqrt {b} E\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )+\left (\sqrt {a}+i \sqrt {b}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right )\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}} \left (-i \sqrt {b} \tan ^2(c+d x)+\sqrt {a} \left (1+\tan ^2(c+d x)\right )\right )}{(a+b) \sqrt {-\frac {i \left (a+i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}+\frac {2 a \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {-\frac {i \left (a+i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}} \left (i \sqrt {b} \tan ^2(c+d x)+\sqrt {a} \left (1+\tan ^2(c+d x)\right )\right )}{\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}+\frac {i a \sqrt {b} \operatorname {EllipticPi}\left (-\frac {2 i \sqrt {b}}{\sqrt {a}-i \sqrt {b}},\arcsin \left (\frac {\sqrt {\frac {i \left (a-i \sqrt {a} \sqrt {b}+a \tan ^2(c+d x)+b \tan ^2(c+d x)\right )}{\sqrt {a} \sqrt {b}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b}}\right ) \sqrt {\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \left (1+\tan ^2(c+d x)\right )}{\sqrt {b}}} \sqrt {\frac {(a+b) \left (b \tan ^4(c+d x)+a \left (1+\tan ^2(c+d x)\right )^2\right )}{a b}}}{\sqrt {a}-i \sqrt {b}}\right )}{2 a d \sqrt {\sec ^2(c+d x)} \left (1+\tan ^2(c+d x)\right )^{3/2} \sqrt {\frac {b \tan ^4(c+d x)+a \left (1+\tan ^2(c+d x)\right )^2}{\left (1+\tan ^2(c+d x)\right )^2}}} \]

[In]

Integrate[Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]

[Out]

-1/4*(Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]]*Cot[c + d*x]*Csc[c + d*x])/(Sqrt[2]*a*d) + (
Sec[c + d*x]*(-a - 2*a*Tan[c + d*x]^2 - a*Tan[c + d*x]^4 - b*Tan[c + d*x]^4 - (b*(Sqrt[b]*EllipticE[ArcSin[Sqr
t[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(
Sqrt[a] - I*Sqrt[b])] + I*(Sqrt[a] + I*Sqrt[b])*EllipticF[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*
x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])])*Sqrt[(((-I)*Sqrt[a]
+ Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^
2))/(Sqrt[a]*Sqrt[b])]*(Sqrt[b]*Tan[c + d*x]^2 + I*Sqrt[a]*(1 + Tan[c + d*x]^2)))/((a + b)*Sqrt[((-I)*(a + I*S
qrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]) + (a*((-I)*Sqrt[b]*EllipticE[ArcSin
[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a
])/(Sqrt[a] - I*Sqrt[b])] + (Sqrt[a] + I*Sqrt[b])*EllipticF[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c +
d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])])*Sqrt[(((-I)*Sqrt[
a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x
]^2))/(Sqrt[a]*Sqrt[b])]*((-I)*Sqrt[b]*Tan[c + d*x]^2 + Sqrt[a]*(1 + Tan[c + d*x]^2)))/((a + b)*Sqrt[((-I)*(a
+ I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]) + (2*a*EllipticF[ArcSin[Sqrt[(
I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqr
t[a] - I*Sqrt[b])]*Sqrt[(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqrt[((-I)*(a + I*Sqrt[a]*Sqr
t[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]*(I*Sqrt[b]*Tan[c + d*x]^2 + Sqrt[a]*(1 + Tan[c
+ d*x]^2)))/((Sqrt[a] + I*Sqrt[b])*Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sq
rt[a]*Sqrt[b])]) + (I*a*Sqrt[b]*EllipticPi[((-2*I)*Sqrt[b])/(Sqrt[a] - I*Sqrt[b]), ArcSin[Sqrt[(I*(a - I*Sqrt[
a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[
b])]*Sqrt[(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqrt[((a + b)*(b*Tan[c + d*x]^4 + a*(1 + Ta
n[c + d*x]^2)^2))/(a*b)])/(Sqrt[a] - I*Sqrt[b])))/(2*a*d*Sqrt[Sec[c + d*x]^2]*(1 + Tan[c + d*x]^2)^(3/2)*Sqrt[
(b*Tan[c + d*x]^4 + a*(1 + Tan[c + d*x]^2)^2)/(1 + Tan[c + d*x]^2)^2])

Maple [F]

\[\int \frac {\csc ^{3}\left (d x +c \right )}{\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}d x\]

[In]

int(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x)

[Out]

int(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x)

Fricas [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]

[In]

integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="fricas")

[Out]

integral(csc(d*x + c)^3/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)

Sympy [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]

[In]

integrate(csc(d*x+c)**3/(a+b*sin(d*x+c)**4)**(1/2),x)

[Out]

Integral(csc(c + d*x)**3/sqrt(a + b*sin(c + d*x)**4), x)

Maxima [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]

[In]

integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(csc(d*x + c)^3/sqrt(b*sin(d*x + c)^4 + a), x)

Giac [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]

[In]

integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]

[In]

int(1/(sin(c + d*x)^3*(a + b*sin(c + d*x)^4)^(1/2)),x)

[Out]

int(1/(sin(c + d*x)^3*(a + b*sin(c + d*x)^4)^(1/2)), x)