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

Optimal. Leaf size=861 \[ -\frac{\sqrt{a-b+c} \tan ^{-1}\left (\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right )}{2 e}-\frac{\cot (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e}+\frac{\sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e \left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )}-\frac{\sqrt [4]{a} \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{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\sqrt [4]{c} (a-b+c) \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 ) \left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt{a}-\sqrt{c}\right ) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\left (\sqrt{a}+\sqrt{c}\right ) \sqrt [4]{c} \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 ) \left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{2 \sqrt [4]{a} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left (\sqrt{a}+\sqrt{c}\right ) (a-b+c) \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{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt{a}-\sqrt{c}\right ) \sqrt [4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}} \]

[Out]

-(Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e)
- (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/e + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*
x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*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])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x
]^4]) + ((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] + Sqrt[c]*T
an[d + e*x]^2)^2])/(2*a^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*(a - b + c)*Elliptic
F[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)*Sq
rt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt
[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*(a - b + 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 = 0.585023, antiderivative size = 861, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3700, 1311, 1281, 1197, 1103, 1195, 1216, 1706} \[ -\frac{\sqrt{a-b+c} \tan ^{-1}\left (\frac{\sqrt{a-b+c} \tan (d+e x)}{\sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right )}{2 e}-\frac{\cot (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e}+\frac{\sqrt{c} \tan (d+e x) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e \left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )}-\frac{\sqrt [4]{a} \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{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\sqrt [4]{c} (a-b+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{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt{a}-\sqrt{c}\right ) e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac{\left (\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{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{2 \sqrt [4]{a} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac{\left (\sqrt{a}+\sqrt{c}\right ) (a-b+c) \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{c} \tan ^2(d+e x)+\sqrt{a}\right ) \sqrt{\frac{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt{c} \tan ^2(d+e x)+\sqrt{a}\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt{a}-\sqrt{c}\right ) \sqrt [4]{c} e \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}} \]

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]

-(Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e)
- (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/e + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*
x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*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])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x
]^4]) + ((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] + Sqrt[c]*T
an[d + e*x]^2)^2])/(2*a^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*(a - b + c)*Elliptic
F[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)*Sq
rt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt
[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*(a - b + 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 1311

Int[(((f_.)*(x_))^(m_)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[
1/(d*e), Int[(f*x)^m*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] - Dist[(c*d^2 - b*d*e + a*e^2)/(d*e*f
^2), Int[((f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Ne
Q[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, 0]

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*(
f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1197

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

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 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 1216

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Di
st[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(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}, 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]

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 \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{\sqrt{a+b x^2+c x^4}}{x^2 \left (1+x^2\right )} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+c x^2}{x^2 \sqrt{a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e}-\frac{(a-b+c) \operatorname{Subst}\left (\int \frac{1}{\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)}}{e}-\frac{\operatorname{Subst}\left (\int \frac{-a c-a c x^2}{\sqrt{a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{a e}-\frac{\left (\sqrt{a} (a-b+c)\right ) \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 (\sqrt{c} (a-b+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{\sqrt{a-b+c} \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}-\frac{\cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e}+\frac{\sqrt [4]{c} (a-b+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 \sqrt [4]{a} \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 ) (a-b+c) \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)}}-\frac{\left (\sqrt{a} \sqrt{c}\right ) \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 )}{e}+\frac{\left (\left (\sqrt{a}+\sqrt{c}\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 )}{e}\\ &=-\frac{\sqrt{a-b+c} \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}-\frac{\cot (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e}+\frac{\sqrt{c} \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e \left (\sqrt{a}+\sqrt{c} \tan ^2(d+e x)\right )}-\frac{\sqrt [4]{a} \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}}}{e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\left (\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 \sqrt [4]{a} e \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac{\sqrt [4]{c} (a-b+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 \sqrt [4]{a} \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 ) (a-b+c) \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 = 26.9031, size = 1258, normalized size = 1.46 \[ \frac{\sqrt{\frac{4 \cos (2 (d+e x)) a+\cos (4 (d+e x)) a+3 a+b+3 c-4 c \cos (2 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{4 \cos (2 (d+e x))+\cos (4 (d+e x))+3}} \left (\frac{1}{2} \sin (2 (d+e x))-\cot (d+e x)\right )}{e}+\frac{i \sqrt{2} \left (\sqrt{b^2-4 a c}-b\right ) \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}{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{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left (\tan ^2(d+e x)+1\right )-2 i \sqrt{2} c \text{EllipticF}\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 ) \sqrt{\frac{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left (\tan ^2(d+e x)+1\right )+2 i \sqrt{2} a \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{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left (\tan ^2(d+e x)+1\right )-2 i \sqrt{2} b \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{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left (\tan ^2(d+e x)+1\right )+2 i \sqrt{2} c \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{2 c \tan ^2(d+e x)+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c \tan ^2(d+e x)}{b-\sqrt{b^2-4 a c}}+1} \left (\tan ^2(d+e x)+1\right )-4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} \tan (d+e x) \left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )}{4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e \left (\tan ^2(d+e x)+1\right ) \sqrt{c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}} \]

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] + Sin[2*(d + e*x)]/2))/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])])*(1 + Tan[d + e*x]^2)*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])] - (
2*I)*Sqrt[2]*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])]*(1 + Tan[d + e*x]^2)*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])] + (2*I)*Sqrt[2]*a*EllipticPi[(b + Sqrt[b
^2 - 4*a*c])/(2*c), 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])]*(1 + Tan[d + e*x]^2)*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])] - (2*I)*Sqrt[2]*b*EllipticPi[(b + Sqrt[b^2 -
4*a*c])/(2*c), 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 - S
qrt[b^2 - 4*a*c])]*(1 + Tan[d + e*x]^2)*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])] + (2*I)*Sqrt[2]*c*EllipticPi[(b + Sqrt[b^2 - 4*a*c
])/(2*c), 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])]*(1 + Tan[d + e*x]^2)*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])] - 4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]*(a + b
*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*(1 + Tan[d + e*x]^2)*Sqrt[a + b*Tan[
d + e*x]^2 + c*Tan[d + e*x]^4])

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Maple [F]  time = 0.497, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( ex+d \right ) \right ) ^{2}\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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{2}\,{d x} \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]

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

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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 \sqrt{a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \cot ^{2}{\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(sqrt(a + b*tan(d + e*x)**2 + c*tan(d + e*x)**4)*cot(d + e*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{2}\,{d x} \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]

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