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

Optimal. Leaf size=707 \[ \frac{\sqrt [4]{c} \left (2 \sqrt{a}-\sqrt{c}\right ) \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right ) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{3/4} e \left (\sqrt{a}-\sqrt{c}\right ) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right ) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{3/4} e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt{a-b+c}}+\frac{\sqrt{c} \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )}-\frac{\cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e}-\frac{\left (\sqrt{a}+\sqrt{c}\right ) \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right ) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )^2}} \Pi \left (-\frac{\left (\sqrt{a}-\sqrt{c}\right )^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}-\sqrt{c}\right ) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

[Out]

-ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*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)*Ta
n[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] + Sqr
t[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]) - ((S
qrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*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])

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Rubi [A]  time = 0.727519, antiderivative size = 707, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3700, 1329, 1714, 1195, 1708, 1103, 1706} \[ \frac{\sqrt [4]{c} \left (2 \sqrt{a}-\sqrt{c}\right ) \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right ) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{3/4} e \left (\sqrt{a}-\sqrt{c}\right ) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right ) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{3/4} e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt{a-b+c}}+\frac{\sqrt{c} \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )}-\frac{\cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{a e}-\frac{\left (\sqrt{a}+\sqrt{c}\right ) \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right ) \sqrt{\frac{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )^2}} \Pi \left (-\frac{\left (\sqrt{a}-\sqrt{c}\right )^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}-\sqrt{c}\right ) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*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)*Ta
n[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] + Sqr
t[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]) - ((S
qrt[a] + Sqrt[c])*EllipticPi[-(Sqrt[a] - Sqrt[c])^2/(4*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 3700

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]

Rule 1329

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)*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])/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; Fr
eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[m/2, 0]

Rule 1714

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 1195

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)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), 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 1708

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 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1706

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)]*EllipticPi[Cancel[-((B*d - A*e)^2/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2 - (b*A)/(4*a*B)])/(4*d*e*A*q*Sqrt[
a + b*x^2 + c*x^4]), 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]

Rubi steps

\begin{align*} \int \frac{\cot ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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} \operatorname{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 \tan ^{-1}\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} \operatorname{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 ) \operatorname{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{\tan ^{-1}\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 \tan ^{-1}\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} F\left (2 \tan ^{-1}\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 ) \Pi \left (-\frac{\left (\sqrt{a}-\sqrt{c}\right )^2}{4 \sqrt{a} \sqrt{c}};2 \tan ^{-1}\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]  time = 27.8916, size = 683, normalized size = 0.97 \[ \frac{\left (\frac{\sin (2 (d+e x))}{2 a}-\frac{\cot (d+e x)}{a}\right ) \sqrt{\frac{4 a \cos (2 (d+e x))+a \cos (4 (d+e x))+3 a-b \cos (4 (d+e x))+b-4 c \cos (2 (d+e x))+c \cos (4 (d+e x))+3 c}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}}}{e}+\frac{\frac{i \sqrt{2} \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \tan (d+e x)\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}+\frac{2 i \sqrt{2} a \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c \tan ^2(d+e x)}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \Pi \left (\frac{b+\sqrt{b^2-4 a c}}{2 c};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x)\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}-\frac{4 \tan (d+e x) \left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )}{\tan ^2(d+e x)+1}}{4 a e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

Warning: Unable to verify antiderivative.

[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])

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Maple [F]  time = 0.537, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( ex+d \right ) \right ) ^{2}{\frac{1}{\sqrt{a+b \left ( \tan \left ( ex+d \right ) \right ) ^{2}+c \left ( \tan \left ( ex+d \right ) \right ) ^{4}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out