3.687 \(\int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\)

Optimal. Leaf size=546 \[ -\frac{d \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} f}-\frac{d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac{d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac{\sqrt{3} d \sqrt [3]{c-\sqrt{-d^2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt{-d^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-d^2} f}+\frac{\sqrt{3} d \sqrt [3]{c+\sqrt{-d^2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt{-d^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-d^2} f}+\frac{3 d \sqrt [3]{c-\sqrt{-d^2}} \log \left (\sqrt [3]{c-\sqrt{-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt{-d^2} f}-\frac{3 d \sqrt [3]{c+\sqrt{-d^2}} \log \left (\sqrt [3]{c+\sqrt{-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt{-d^2} f}+\frac{d \sqrt [3]{c-\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}-\frac{d \sqrt [3]{c+\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}+\frac{1}{4} x \sqrt [3]{c-\sqrt{-d^2}}+\frac{1}{4} x \sqrt [3]{c+\sqrt{-d^2}}-\frac{\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f} \]

[Out]

((c - Sqrt[-d^2])^(1/3)*x)/4 + ((c + Sqrt[-d^2])^(1/3)*x)/4 - (d*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3
))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*f) - (Sqrt[3]*d*(c - Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f
*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (Sqrt[3]*d*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 +
(2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (d*(c - Sqrt[-d^2])^(1/3)
*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*Log[
Tan[e + f*x]])/(6*c^(2/3)*f) + (d*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(2*c^(2/3)*f) + (3*d*(c - Sqrt[-d
^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (3*d*(c + Sqrt[-d^2])^
(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (Cot[e + f*x]*(c + d*Tan[e
+ f*x])^(1/3))/f

________________________________________________________________________________________

Rubi [A]  time = 0.581506, antiderivative size = 546, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3568, 3653, 3485, 712, 50, 57, 617, 204, 31, 3634} \[ -\frac{d \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} f}-\frac{d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac{d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac{\sqrt{3} d \sqrt [3]{c-\sqrt{-d^2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt{-d^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-d^2} f}+\frac{\sqrt{3} d \sqrt [3]{c+\sqrt{-d^2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt{-d^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-d^2} f}+\frac{3 d \sqrt [3]{c-\sqrt{-d^2}} \log \left (\sqrt [3]{c-\sqrt{-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt{-d^2} f}-\frac{3 d \sqrt [3]{c+\sqrt{-d^2}} \log \left (\sqrt [3]{c+\sqrt{-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt{-d^2} f}+\frac{d \sqrt [3]{c-\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}-\frac{d \sqrt [3]{c+\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}+\frac{1}{4} x \sqrt [3]{c-\sqrt{-d^2}}+\frac{1}{4} x \sqrt [3]{c+\sqrt{-d^2}}-\frac{\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3),x]

[Out]

((c - Sqrt[-d^2])^(1/3)*x)/4 + ((c + Sqrt[-d^2])^(1/3)*x)/4 - (d*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3
))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*f) - (Sqrt[3]*d*(c - Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f
*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (Sqrt[3]*d*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 +
(2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (d*(c - Sqrt[-d^2])^(1/3)
*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*Log[
Tan[e + f*x]])/(6*c^(2/3)*f) + (d*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(2*c^(2/3)*f) + (3*d*(c - Sqrt[-d
^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (3*d*(c + Sqrt[-d^2])^
(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (Cot[e + f*x]*(c + d*Tan[e
+ f*x])^(1/3))/f

Rule 3568

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rule 3485

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 712

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

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

Mathematica [C]  time = 3.22557, size = 464, normalized size = 0.85 \[ \frac{-\frac{1}{6} \sqrt [3]{c} d \left (\log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )\right )+d \sqrt [3]{c+d \tan (e+f x)}+\frac{1}{4} i c \sqrt [3]{c-i d} \left (2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )-2 \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )+\log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )\right )-\frac{1}{4} i c \sqrt [3]{c+i d} \left (2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )-2 \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )+\log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )\right )+\frac{1}{3} \sqrt [3]{c} d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )-\cot (e+f x) (c+d \tan (e+f x))^{4/3}}{c f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3),x]

[Out]

((c^(1/3)*d*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/3 - (c^(1/3)*d*(2*Sqrt[3]*ArcTan[(c^(1/3) + 2*(c + d*Ta
n[e + f*x])^(1/3))/(Sqrt[3]*c^(1/3))] + Log[c^(2/3) + c^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x]
)^(2/3)]))/6 + (I/4)*c*(c - I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/
Sqrt[3]] - 2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*
Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) - (I/4)*c*(c + I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*
Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c
+ I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) + d*(c + d*Tan[e + f*
x])^(1/3) - Cot[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(c*f)

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{2}\sqrt [3]{c+d\tan \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x)

[Out]

int(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67399, size = 62, normalized size = 0.11 \begin{align*} -\frac{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}{f \tan \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

-(d*tan(f*x + e) + c)^(1/3)/(f*tan(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{c + d \tan{\left (e + f x \right )}} \cot ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2*(c+d*tan(f*x+e))**(1/3),x)

[Out]

Integral((c + d*tan(e + f*x))**(1/3)*cot(e + f*x)**2, x)

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Giac [C]  time = 3.5402, size = 510, normalized size = 0.93 \begin{align*} -\frac{1}{24} \,{\left ({\left (i \, \sqrt{3} + 1\right )} \left (\frac{216 i \, c + 216 \, d}{d^{9} f^{3}}\right )^{\frac{1}{3}} \log \left (d^{2} f\right ) +{\left (-i \, \sqrt{3} + 1\right )} \left (\frac{216 i \, c + 216 \, d}{d^{9} f^{3}}\right )^{\frac{1}{3}} \log \left (d^{2} f\right ) +{\left (i \, \sqrt{3} + 1\right )} \left (\frac{-216 i \, c + 216 \, d}{d^{9} f^{3}}\right )^{\frac{1}{3}} \log \left (d^{2} f\right ) +{\left (-i \, \sqrt{3} + 1\right )} \left (\frac{-216 i \, c + 216 \, d}{d^{9} f^{3}}\right )^{\frac{1}{3}} \log \left (d^{2} f\right ) - 2 \, \left (\frac{216 i \, c + 216 \, d}{d^{9} f^{3}}\right )^{\frac{1}{3}} \log \left (i \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{2} f +{\left (i \, c + d\right )}^{\frac{1}{3}} d^{2} f\right ) - 2 \, \left (\frac{-216 i \, c + 216 \, d}{d^{9} f^{3}}\right )^{\frac{1}{3}} \log \left (-i \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{2} f +{\left (-i \, c + d\right )}^{\frac{1}{3}} d^{2} f\right ) + \frac{8 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} + c^{\frac{1}{3}}\right )}}{3 \, c^{\frac{1}{3}}}\right )}{c^{\frac{2}{3}} d^{2} f} + \frac{4 \, \log \left ({\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{2}{3}} +{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} c^{\frac{1}{3}} + c^{\frac{2}{3}}\right )}{c^{\frac{2}{3}} d^{2} f} - \frac{8 \, \log \left ({\left |{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} - c^{\frac{1}{3}} \right |}\right )}{c^{\frac{2}{3}} d^{2} f} + \frac{24 \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}{d^{3} f \tan \left (f x + e\right )}\right )} d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x, algorithm="giac")

[Out]

-1/24*((I*sqrt(3) + 1)*((216*I*c + 216*d)/(d^9*f^3))^(1/3)*log(d^2*f) + (-I*sqrt(3) + 1)*((216*I*c + 216*d)/(d
^9*f^3))^(1/3)*log(d^2*f) + (I*sqrt(3) + 1)*((-216*I*c + 216*d)/(d^9*f^3))^(1/3)*log(d^2*f) + (-I*sqrt(3) + 1)
*((-216*I*c + 216*d)/(d^9*f^3))^(1/3)*log(d^2*f) - 2*((216*I*c + 216*d)/(d^9*f^3))^(1/3)*log(I*(d*tan(f*x + e)
+ c)^(1/3)*d^2*f + (I*c + d)^(1/3)*d^2*f) - 2*((-216*I*c + 216*d)/(d^9*f^3))^(1/3)*log(-I*(d*tan(f*x + e) + c
)^(1/3)*d^2*f + (-I*c + d)^(1/3)*d^2*f) + 8*sqrt(3)*arctan(1/3*sqrt(3)*(2*(d*tan(f*x + e) + c)^(1/3) + c^(1/3)
)/c^(1/3))/(c^(2/3)*d^2*f) + 4*log((d*tan(f*x + e) + c)^(2/3) + (d*tan(f*x + e) + c)^(1/3)*c^(1/3) + c^(2/3))/
(c^(2/3)*d^2*f) - 8*log(abs((d*tan(f*x + e) + c)^(1/3) - c^(1/3)))/(c^(2/3)*d^2*f) + 24*(d*tan(f*x + e) + c)^(
1/3)/(d^3*f*tan(f*x + e)))*d^3