\(\int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx\) [44]

Optimal result

Integrand size = 35, antiderivative size = 707 \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}-\frac {\cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e}+\frac {\sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}-\frac {\sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{a^{3/4} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (2 \sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 a^{3/4} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

[Out]

-1/2*arctan((a-b+c)^(1/2)*tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))/e/(a-b+c)^(1/2)-cot(e*x+d)*(a+b*
tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/a/e+c^(1/2)*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)/a/e/(a^(1/2)
+c^(1/2)*tan(e*x+d)^2)-c^(1/4)*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*tan(e*
x+d)/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*((a+b*tan(
e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/a^(3/4)/e/(a+b
*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)+1/2*c^(1/4)*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arct
an(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^
(1/2))*(2*a^(1/2)-c^(1/2))*((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)
+c^(1/2)*tan(e*x+d)^2)/a^(3/4)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-1/4*(cos(2*arctan(c
^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)
*tan(e*x+d)/a^(1/4))),-1/4*(a^(1/2)-c^(1/2))^2/a^(1/2)/c^(1/2),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+c^(1/
2))*((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/
a^(1/4)/c^(1/4)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3781, 1343, 1728, 1209, 1722, 1117, 1720} \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\frac {\sqrt [4]{c} \left (2 \sqrt {a}-\sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} e \left (\sqrt {a}-\sqrt {c}\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}-\sqrt {c}\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}-\frac {\cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e} \]

[In]

Int[Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]

[Out]

-1/2*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(Sqrt[a - b + c]*e)
- (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a*e) + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d
+ e*x]^2 + c*Tan[d + e*x]^4])/(a*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*
Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*
x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d
+ e*x]^4]) + ((2*Sqrt[a] - Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[
a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + S
qrt[c]*Tan[d + e*x]^2)^2])/(2*a^(3/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (
(Sqrt[a] + Sqrt[c])*EllipticPi[-1/4*(Sqrt[a] - Sqrt[c])^2/(Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a
^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d
+ e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e
*x]^2 + c*Tan[d + e*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 1343

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

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 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)
), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.21 (sec) , antiderivative size = 683, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\frac {\sqrt {\frac {3 a+b+3 c+4 a \cos (2 (d+e x))-4 c \cos (2 (d+e x))+a \cos (4 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{3+4 \cos (2 (d+e x))+\cos (4 (d+e x))}} \left (-\frac {\cot (d+e x)}{a}+\frac {\sin (2 (d+e x))}{2 a}\right )}{e}+\frac {\frac {i \sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}+\frac {2 i \sqrt {2} a \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}-\frac {4 \tan (d+e x) \left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )}{1+\tan ^2(d+e x)}}{4 a e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

[In]

Integrate[Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]

[Out]

(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos[4*(d + e*x)] - b*Cos[4*(d + e*x)] +
c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(-(Cot[d + e*x]/a) + Sin[2*(d + e*x)]/(2*a))
)/e + ((I*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d
+ e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 -
4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*T
an[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sq
rt[b^2 - 4*a*c])] + ((2*I)*Sqrt[2]*a*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + S
qrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c
/(b + Sqrt[b^2 - 4*a*c])] - (4*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(1 + Tan[d + e*x]^2))/(
4*a*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])

Maple [F]

\[\int \frac {\cot \left (e x +d \right )^{2}}{\sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}d x\]

[In]

int(cot(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x)

[Out]

int(cot(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\text {Timed out} \]

[In]

integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int \frac {\cot ^{2}{\left (d + e x \right )}}{\sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}}}\, dx \]

[In]

integrate(cot(e*x+d)**2/(a+b*tan(e*x+d)**2+c*tan(e*x+d)**4)**(1/2),x)

[Out]

Integral(cot(d + e*x)**2/sqrt(a + b*tan(d + e*x)**2 + c*tan(d + e*x)**4), x)

Maxima [F]

\[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a}} \,d x } \]

[In]

integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(e*x + d)^2/sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a), x)

Giac [F]

\[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a}} \,d x } \]

[In]

integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^2}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a}} \,d x \]

[In]

int(cot(d + e*x)^2/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(1/2),x)

[Out]

int(cot(d + e*x)^2/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(1/2), x)