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

Optimal. Leaf size=402 \[ -\frac{\sqrt{3} \sqrt [3]{c} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{f}+\frac{\sqrt{3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{2 f}+\frac{\sqrt{3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{2 f}+\frac{3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac{3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac{3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt [3]{c-i d}+\frac{1}{4} i x \sqrt [3]{c+i d}-\frac{\sqrt [3]{c} \log (\tan (e+f x))}{2 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.477561, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {3574, 3528, 3539, 3537, 57, 617, 204, 31, 3634, 50} \[ -\frac{\sqrt{3} \sqrt [3]{c} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{f}+\frac{\sqrt{3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{2 f}+\frac{\sqrt{3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{2 f}+\frac{3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac{3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac{3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt [3]{c-i d}+\frac{1}{4} i x \sqrt [3]{c+i d}-\frac{\sqrt [3]{c} \log (\tan (e+f x))}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3574

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

Rule 3528

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

Rule 3539

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

Rule 3537

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.784554, size = 744, normalized size = 1.85 \[ \frac{-2 \sqrt [3]{c} \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )-4 \sqrt{3} \sqrt [3]{c} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )+2 \sqrt{3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )+\frac{2 i \sqrt{3} d \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{(c+i d)^{2/3}}+\frac{2 \sqrt{3} c \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{(c+i d)^{2/3}}+4 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )+\frac{2 i d \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{(c-i d)^{2/3}}-\frac{2 c \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{(c-i d)^{2/3}}-\frac{2 i d \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{(c+i d)^{2/3}}-\frac{2 c \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{(c+i d)^{2/3}}-\frac{i d \log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )}{(c-i d)^{2/3}}+\frac{c \log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )}{(c-i d)^{2/3}}+\frac{i d \log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )}{(c+i d)^{2/3}}+\frac{c \log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )}{(c+i d)^{2/3}}}{4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( fx+e \right ) \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)*(c+d*tan(f*x+e))^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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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{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} \cot \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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