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

Optimal. Leaf size=943 \[ -\frac {\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a} \cot ^3(d+e x)}{3 e}+\frac {(3 a-b) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a} \cot (d+e x)}{3 a e}+\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 {(3 a-b) \sqrt {c} \tan (d+e x) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{3 a e \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )}+\frac {(3 a-b) \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}}}{3 a^{3/4} 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 (3 a+\sqrt {c} \sqrt {a}-b\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}}}{6 a^{3/4} 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]

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))*(a-b+c)^(1/2)/e+1/3*(3*a-b)*cot(e
*x+d)*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/a/e-1/3*cot(e*x+d)^3*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/e-1
/3*(3*a-b)*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)+1/3*(
3*a-b)*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)))*E
llipticE(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)*(a-b+c)*(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)))*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))*(
(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)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)+1/4*(a-b+c)*(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)-1/6*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)))*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))*(3*a-b+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+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.72, antiderivative size = 943, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3700, 1309, 1281, 1197, 1103, 1195, 1216, 1706} \[ -\frac {\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a} \cot ^3(d+e x)}{3 e}+\frac {(3 a-b) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a} \cot (d+e x)}{3 a e}+\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 {(3 a-b) \sqrt {c} \tan (d+e x) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{3 a e \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )}+\frac {(3 a-b) \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}}}{3 a^{3/4} 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 (3 a+\sqrt {c} \sqrt {a}-b\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}}}{6 a^{3/4} 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]^4*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) +
((3*a - b)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*a*e) - (Cot[d + e*x]^3*Sqrt[a + b*T
an[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*e) - ((3*a - b)*Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d
+ e*x]^4])/(3*a*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + ((3*a - b)*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])/(3*a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e
*x]^4]) - ((3*a - b + Sqrt[a]*Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqr
t[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])/(6*a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (c^(1/4)*(a - b + c
)*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]*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 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 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 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 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 1309

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

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]

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]

Rubi steps

\begin {align*} \int \cot ^4(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^4 \left (1+x^2\right )} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+(-a+b) x^2}{x^4 \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 ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 e}-\frac {\operatorname {Subst}\left (\int \frac {a (3 a-b)+a c x^2}{x^2 \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{3 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 {(3 a-b) \cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 a e}-\frac {\cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 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 {\operatorname {Subst}\left (\int \frac {-a^2 c-a (3 a-b) c x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{3 a^2 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 {(3 a-b) \cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 a e}-\frac {\cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 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 ((3 a-b) \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 )}{3 \sqrt {a} e}-\frac {\left (\left (3 a-b+\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 )}{3 \sqrt {a} 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 {(3 a-b) \cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 a e}-\frac {\cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 e}-\frac {(3 a-b) \sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{3 a e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}+\frac {(3 a-b) \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}}}{3 a^{3/4} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (3 a-b+\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}}}{6 a^{3/4} 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*}

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Mathematica [C]  time = 31.42, size = 1590, normalized size = 1.69 \[ \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}{3} \cot (d+e x) \csc ^2(d+e x)+\frac {(4 a \cos (d+e x)-b \cos (d+e x)) \csc (d+e x)}{3 a}-\frac {(3 a-b) \sin (2 (d+e x))}{6 a}\right )}{e}+\frac {-6 i \sqrt {2} \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 ) \left (\tan ^2(d+e x)+1\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} a^2+3 i \sqrt {2} \left (b-\sqrt {b^2-4 a c}\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 )-F\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 ) \left (\tan ^2(d+e x)+1\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} a+2 i \sqrt {2} c F\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 ) \left (\tan ^2(d+e x)+1\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} a+6 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 ) \left (\tan ^2(d+e x)+1\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} a-6 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 ) \left (\tan ^2(d+e x)+1\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} a-4 (b-3 a) \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 )+i \sqrt {2} b \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 )-F\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 ) \left (\tan ^2(d+e x)+1\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}}{12 a \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]^4*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)])]*(((4*a*Cos[d + e*x] - b*Cos[d + e*x])*Csc[d
+ e*x])/(3*a) - (Cot[d + e*x]*Csc[d + e*x]^2)/3 - ((3*a - b)*Sin[2*(d + e*x)])/(6*a)))/e + ((3*I)*Sqrt[2]*a*(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*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])] + I*Sqrt[2]*b*(
-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]*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])] - (6*I)*Sqrt[2]*a^2*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 - Sq
rt[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])] + (6*I)*Sqrt[2]*a*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 - 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])] - (6*I)*Sqrt[2]*a*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])]*Sq
rt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])] - 4*(-3*a + b)*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))/(12*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*(1 + Tan[d + e*x]^2)*Sqr
t[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)^4*(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)^4*(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)^4, x)

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maple [F]  time = 1.32, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{4}\left (e x +d \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)^4*(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)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (d+e\,x\right )}^4\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \cot ^{4}{\left (d + e x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)**4*(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)**4, x)

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