\(\int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\) [686]

Optimal result

Integrand size = 21, antiderivative size = 402 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt {3} \sqrt [3]{c} \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{f}+\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f} \]

[Out]

-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-1/4*(c-I*d)^(1/3)*ln(cos(f*x+e))/f-1/4*(c+I*d)^(1/3)*ln(cos(f*x+e
))/f-1/2*c^(1/3)*ln(tan(f*x+e))/f+3/2*c^(1/3)*ln(c^(1/3)-(c+d*tan(f*x+e))^(1/3))/f-3/4*(c-I*d)^(1/3)*ln((c-I*d
)^(1/3)-(c+d*tan(f*x+e))^(1/3))/f-3/4*(c+I*d)^(1/3)*ln((c+I*d)^(1/3)-(c+d*tan(f*x+e))^(1/3))/f-c^(1/3)*arctan(
1/3*(c^(1/3)+2*(c+d*tan(f*x+e))^(1/3))/c^(1/3)*3^(1/2))*3^(1/2)/f+1/2*(c-I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan(f
*x+e))^(1/3)/(c-I*d)^(1/3))*3^(1/2))*3^(1/2)/f+1/2*(c+I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c+I*d
)^(1/3))*3^(1/2))*3^(1/2)/f

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3655, 3609, 3620, 3618, 59, 631, 210, 31, 3715, 52} \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{c} \arctan \left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{f}+\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {1}{4} i x \sqrt [3]{c-i d}+\frac {1}{4} i x \sqrt [3]{c+i d}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f} \]

[In]

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

[Out]

(-1/4*I)*(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 31

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

Rule 52

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 59

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 210

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

Rule 631

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 3609

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 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 3655

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 3715

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*} \text {integral}& = -\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 {\text {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 \text {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 ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \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 ) \text {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 ) \text {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} \arctan \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} \arctan \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} \arctan \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] (verified)

Time = 0.94 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.85 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {-4 \sqrt {3} \sqrt [3]{c} \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )+2 \sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )+\frac {2 \sqrt {3} c \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{(c+i d)^{2/3}}+\frac {2 i \sqrt {3} d \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{(c+i d)^{2/3}}+4 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )-\frac {2 c \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c-i d)^{2/3}}+\frac {2 i d \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c-i d)^{2/3}}-\frac {2 c \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c+i d)^{2/3}}-\frac {2 i d \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c+i d)^{2/3}}-2 \sqrt [3]{c} \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+\frac {c \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c-i d)^{2/3}}-\frac {i d \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c-i d)^{2/3}}+\frac {c \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c+i d)^{2/3}}+\frac {i d \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c+i d)^{2/3}}}{4 f} \]

[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)

Maple [F]

\[\int \cot \left (f x +e \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.40 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {{\left (\sqrt {-3} f + f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} f + f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} f - f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} f - f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 2 \, f \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (f \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + {\left (\sqrt {-3} f + f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} f + f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} f - f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} f - f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 2 \, f \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (f \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + 4 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} c^{\frac {2}{3}} + \sqrt {3} c}{3 \, c}\right ) + 2 \, c^{\frac {1}{3}} \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 ) - 4 \, c^{\frac {1}{3}} \log \left ({\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} - c^{\frac {1}{3}}\right )}{4 \, f} \]

[In]

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

[Out]

-1/4*((sqrt(-3)*f + f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(-1/2*(sqrt(-3)*f + f)*(-(f^3*sqrt(-d^2/f^6) +
c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) - (sqrt(-3)*f - f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(1/2*
(sqrt(-3)*f - f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) - 2*f*(-(f^3*sqrt(-d^2/f^
6) + c)/f^3)^(1/3)*log(f*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + (sqrt(-3)*f + f
)*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(-1/2*(sqrt(-3)*f + f)*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3) + (d*tan
(f*x + e) + c)^(1/3)) - (sqrt(-3)*f - f)*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(1/2*(sqrt(-3)*f - f)*((f^3*s
qrt(-d^2/f^6) - c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) - 2*f*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(f*(
(f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + 4*sqrt(3)*c^(1/3)*arctan(1/3*(2*sqrt(3)*(d
*tan(f*x + e) + c)^(1/3)*c^(2/3) + sqrt(3)*c)/c) + 2*c^(1/3)*log((d*tan(f*x + e) + c)^(2/3) + (d*tan(f*x + e)
+ c)^(1/3)*c^(1/3) + c^(2/3)) - 4*c^(1/3)*log((d*tan(f*x + e) + c)^(1/3) - c^(1/3)))/f

Sympy [F]

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

[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)

Maxima [F]

\[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right ) \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 17.59 (sec) , antiderivative size = 2133, normalized size of antiderivative = 5.31 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

log((c + d*tan(e + f*x))^(1/3) - f*(c/f^3)^(1/3))*(c/f^3)^(1/3) + log(((-(c - d*1i)/f^3)^(1/3)*(((-(c - d*1i)/
f^3)^(2/3)*(((((104976*c*d^14*(-(c - d*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2) - (104976*c*d^14*(c + d*tan(
e + f*x))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2))/f)*(-(c - d*1i)/f^3)^(2/3))/4 - (78732*c^2*d^14*(c^4 - d^4))/f^3)
*(-(c - d*1i)/f^3)^(1/3))/2 - (39366*c^2*d^14*(c + d*tan(e + f*x))^(1/3)*(5*c^4 + 3*d^4 + 8*c^2*d^2))/f^4))/4
+ (6561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^4*d^2))/f^6))/2 - (6561*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^6
+ d^6 + c^2*d^4 + 3*c^4*d^2))/f^7)*(-(c - d*1i)/(8*f^3))^(1/3) + log(((-(c + d*1i)/f^3)^(1/3)*(((-(c + d*1i)/
f^3)^(2/3)*(((((104976*c*d^14*(-(c + d*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2) - (104976*c*d^14*(c + d*tan(
e + f*x))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2))/f)*(-(c + d*1i)/f^3)^(2/3))/4 - (78732*c^2*d^14*(c^4 - d^4))/f^3)
*(-(c + d*1i)/f^3)^(1/3))/2 - (39366*c^2*d^14*(c + d*tan(e + f*x))^(1/3)*(5*c^4 + 3*d^4 + 8*c^2*d^2))/f^4))/4
+ (6561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^4*d^2))/f^6))/2 - (6561*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^6
+ d^6 + c^2*d^4 + 3*c^4*d^2))/f^7)*(-(c + d*1i)/(8*f^3))^(1/3) + (log(2*(c + d*tan(e + f*x))^(1/3) + f*(c/f^3
)^(1/3) - 3^(1/2)*f*(c/f^3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(c/f^3)^(1/3))/2 - (log(2*(c + d*tan(e + f*x))^(1/3) +
f*(c/f^3)^(1/3) + 3^(1/2)*f*(c/f^3)^(1/3)*1i)*(3^(1/2)*1i + 1)*(c/f^3)^(1/3))/2 + log((((3^(1/2)*1i)/2 - 1/2)*
((((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + 1/2)*(-(c - d*1i)/f^3)^(2/3)*((104976*c*d^14*(c + d*tan(e + f*x
))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2))/f - 104976*c*d^14*((3^(1/2)*1i)/2 - 1/2)*(-(c - d*1i)/f^3)^(1/3)*(3*c^4
+ 2*d^4 + 5*c^2*d^2)))/4 - (78732*c^2*d^14*(c^4 - d^4))/f^3)*(-(c - d*1i)/f^3)^(1/3))/2 - (39366*c^2*d^14*(c +
d*tan(e + f*x))^(1/3)*(5*c^4 + 3*d^4 + 8*c^2*d^2))/f^4)*((3^(1/2)*1i)/2 + 1/2)*(-(c - d*1i)/f^3)^(2/3))/4 - (
6561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^4*d^2))/f^6)*(-(c - d*1i)/f^3)^(1/3))/2 + (6561*c*d^14*(c + d*tan(e
+ f*x))^(1/3)*(3*c^6 + d^6 + c^2*d^4 + 3*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 - 1/2)*(-(c - d*1i)/(8*f^3))^(1/3) +
log((((3^(1/2)*1i)/2 - 1/2)*((((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + 1/2)*(-(c + d*1i)/f^3)^(2/3)*((1049
76*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2))/f - 104976*c*d^14*((3^(1/2)*1i)/2 - 1/2)*(-(
c + d*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2)))/4 - (78732*c^2*d^14*(c^4 - d^4))/f^3)*(-(c + d*1i)/f^3)^(1/
3))/2 - (39366*c^2*d^14*(c + d*tan(e + f*x))^(1/3)*(5*c^4 + 3*d^4 + 8*c^2*d^2))/f^4)*((3^(1/2)*1i)/2 + 1/2)*(-
(c + d*1i)/f^3)^(2/3))/4 - (6561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^4*d^2))/f^6)*(-(c + d*1i)/f^3)^(1/3))/2
+ (6561*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^6 + d^6 + c^2*d^4 + 3*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 - 1/2)*(-
(c + d*1i)/(8*f^3))^(1/3) - log((((3^(1/2)*1i)/2 + 1/2)*((((((3^(1/2)*1i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2)*(-
(c - d*1i)/f^3)^(2/3)*((104976*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2))/f + 104976*c*d^1
4*((3^(1/2)*1i)/2 + 1/2)*(-(c - d*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2)))/4 + (78732*c^2*d^14*(c^4 - d^4)
)/f^3)*(-(c - d*1i)/f^3)^(1/3))/2 - (39366*c^2*d^14*(c + d*tan(e + f*x))^(1/3)*(5*c^4 + 3*d^4 + 8*c^2*d^2))/f^
4)*((3^(1/2)*1i)/2 - 1/2)*(-(c - d*1i)/f^3)^(2/3))/4 + (6561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^4*d^2))/f^6
)*(-(c - d*1i)/f^3)^(1/3))/2 + (6561*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^6 + d^6 + c^2*d^4 + 3*c^4*d^2))/f^
7)*((3^(1/2)*1i)/2 + 1/2)*(-(c - d*1i)/(8*f^3))^(1/3) - log((((3^(1/2)*1i)/2 + 1/2)*((((((3^(1/2)*1i)/2 + 1/2)
*((((3^(1/2)*1i)/2 - 1/2)*(-(c + d*1i)/f^3)^(2/3)*((104976*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^4 + 4*d^4 +
7*c^2*d^2))/f + 104976*c*d^14*((3^(1/2)*1i)/2 + 1/2)*(-(c + d*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2)))/4 +
(78732*c^2*d^14*(c^4 - d^4))/f^3)*(-(c + d*1i)/f^3)^(1/3))/2 - (39366*c^2*d^14*(c + d*tan(e + f*x))^(1/3)*(5*
c^4 + 3*d^4 + 8*c^2*d^2))/f^4)*((3^(1/2)*1i)/2 - 1/2)*(-(c + d*1i)/f^3)^(2/3))/4 + (6561*c*d^14*(d^6 - 3*c^6 +
7*c^2*d^4 + 3*c^4*d^2))/f^6)*(-(c + d*1i)/f^3)^(1/3))/2 + (6561*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^6 + d^
6 + c^2*d^4 + 3*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 + 1/2)*(-(c + d*1i)/(8*f^3))^(1/3)