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]

-1/6*d*ln(tan(f*x+e))/c^(2/3)/f+1/2*d*ln(c^(1/3)-(c+d*tan(f*x+e))^(1/3))/c^(2/3)/f-1/3*d*arctan(1/3*(c^(1/3)+2
*(c+d*tan(f*x+e))^(1/3))/c^(1/3)*3^(1/2))/c^(2/3)/f*3^(1/2)+1/4*x*(c-(-d^2)^(1/2))^(1/3)+1/4*d*ln(cos(f*x+e))*
(c-(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)+3/4*d*ln((c-(-d^2)^(1/2))^(1/3)-(c+d*tan(f*x+e))^(1/3))*(c-(-d^2)^(1/2))
^(1/3)/f/(-d^2)^(1/2)-1/2*d*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c-(-d^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)*(c
-(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)+1/4*x*(c+(-d^2)^(1/2))^(1/3)-1/4*d*ln(cos(f*x+e))*(c+(-d^2)^(1/2))^(1/3)/f
/(-d^2)^(1/2)-3/4*d*ln((c+(-d^2)^(1/2))^(1/3)-(c+d*tan(f*x+e))^(1/3))*(c+(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)+1/
2*d*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c+(-d^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)*(c+(-d^2)^(1/2))^(1/3)/f/(
-d^2)^(1/2)-cot(f*x+e)*(c+d*tan(f*x+e))^(1/3)/f

________________________________________________________________________________________

Rubi [A]  time = 0.58, antiderivative size = 546, normalized size of antiderivative = 1.00, 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 31

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

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 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 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 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 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 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 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 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]

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 = 3.37, 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)

________________________________________________________________________________________

fricas [A]  time = 1.65, size = 25, normalized size = 0.05 \[ -\frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}{f \tan \left (f x + e\right )} \]

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))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]

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]

undef

________________________________________________________________________________________

maple [F]  time = 0.63, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{2}\left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right )^{2}\,{d x} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 21.08, size = 3239, normalized size = 5.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((((((243*(c + d*tan(e + f*x))^(1/3)*(576*d^19*f^6 + 1584*c^2*d^17*f^6 + 1008*c^4*d^15*f^6))/f^7 - (243*(17
28*c*d^18*f^6 + 4320*c^3*d^16*f^6 + 2592*c^5*d^14*f^6)*(d^3/(27*c^2*f^3))^(1/3))/f^6)*(d^3/(27*c^2*f^3))^(2/3)
+ (243*(464*c*d^19*f^3 + 1112*c^3*d^17*f^3 + 648*c^5*d^15*f^3))/f^6)*(d^3/(27*c^2*f^3))^(1/3) - (243*(c + d*t
an(e + f*x))^(1/3)*(160*d^20*f^3 + 358*c^2*d^18*f^3 + 144*c^4*d^16*f^3 - 54*c^6*d^14*f^3))/f^7)*(d^3/(27*c^2*f
^3))^(2/3) - (243*(35*c*d^20 + 89*c^3*d^18 + 81*c^5*d^16 + 27*c^7*d^14))/f^6)*(d^3/(27*c^2*f^3))^(1/3) + (243*
(c + d*tan(e + f*x))^(1/3)*(11*d^21 + 27*c^2*d^19 + 25*c^4*d^17 + 9*c^6*d^15))/f^7)*(d^3/(27*c^2*f^3))^(1/3) +
log((243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7 - (((c*1i + d)/f^3)^
(1/3)*((((((c*1i + d)/f^3)^(1/3)*(((104976*c*d^14*((c*1i + d)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2) - (34992*
d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f)*((c*1i + d)/f^3)^(2/3))/4 - (1944*c*d^15*(81*
c^4 + 58*d^4 + 139*c^2*d^2))/f^3))/2 + (486*d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 7
2*c^4*d^2))/f^4)*((c*1i + d)/f^3)^(2/3))/4 + (243*c*d^14*(27*c^6 + 35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6))/2)
*((c*1i + d)/(8*f^3))^(1/3) + log(((-(c*1i - d)/f^3)^(1/3)*(((-(c*1i - d)/f^3)^(2/3)*(((-(c*1i - d)/f^3)^(1/3)
*(((-(c*1i - d)/f^3)^(2/3)*(104976*c*d^14*(-(c*1i - d)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2) - (34992*d^15*(c
+ d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f))/4 - (1944*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f
^3))/2 + (486*d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 72*c^4*d^2))/f^4))/4 + (243*c*d
^14*(27*c^6 + 35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6))/2 - (243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^
6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7)*(-(c*1i - d)/(8*f^3))^(1/3) + log(- ((-(c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/
2 - 1/2)*(((-(c*1i - d)/f^3)^(2/3)*((3^(1/2)*1i)/2 + 1/2)*(((-(c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(((
-(c*1i - d)/f^3)^(2/3)*((3^(1/2)*1i)/2 + 1/2)*((34992*d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*
d^2))/f - 104976*c*d^14*(-(c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(3*c^4 + 2*d^4 + 5*c^2*d^2)))/4 - (1944
*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f^3))/2 + (486*d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179
*c^2*d^4 + 72*c^4*d^2))/f^4))/4 - (243*c*d^14*(27*c^6 + 35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6))/2 - (243*d^15
*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 - 1/2)*(-(c*1i -
d)/(8*f^3))^(1/3) - log(((-(c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/2 + 1/2)*(((-(c*1i - d)/f^3)^(2/3)*((3^(1/2)*1i
)/2 - 1/2)*(((-(c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/2 + 1/2)*(((-(c*1i - d)/f^3)^(2/3)*((3^(1/2)*1i)/2 - 1/2)*(
(34992*d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f + 104976*c*d^14*(-(c*1i - d)/f^3)^(1/3)
*((3^(1/2)*1i)/2 + 1/2)*(3*c^4 + 2*d^4 + 5*c^2*d^2)))/4 + (1944*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f^3))/
2 + (486*d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 72*c^4*d^2))/f^4))/4 + (243*c*d^14*(
27*c^6 + 35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6))/2 + (243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^6 + 2
7*c^2*d^4 + 25*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 + 1/2)*(-(c*1i - d)/(8*f^3))^(1/3) + (d*(c + d*tan(e + f*x))^(1/
3))/(c*f - f*(c + d*tan(e + f*x))) + log(- (((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + 1/2)*((c*1i + d)/f^3)^(
2/3)*((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + 1/2)*((34992*d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4
+ 11*c^2*d^2))/f - 104976*c*d^14*((3^(1/2)*1i)/2 - 1/2)*((c*1i + d)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2))*((
c*1i + d)/f^3)^(2/3))/4 - (1944*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f^3)*((c*1i + d)/f^3)^(1/3))/2 + (486*
d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 72*c^4*d^2))/f^4))/4 - (243*c*d^14*(27*c^6 +
35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6)*((c*1i + d)/f^3)^(1/3))/2 - (243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^
6 + 11*d^6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 - 1/2)*((c*1i + d)/(8*f^3))^(1/3) - log((((3^(1/2)
*1i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2)*((c*1i + d)/f^3)^(2/3)*((((3^(1/2)*1i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2
)*((34992*d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f + 104976*c*d^14*((3^(1/2)*1i)/2 + 1/
2)*((c*1i + d)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2))*((c*1i + d)/f^3)^(2/3))/4 + (1944*c*d^15*(81*c^4 + 58*d
^4 + 139*c^2*d^2))/f^3)*((c*1i + d)/f^3)^(1/3))/2 + (486*d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 17
9*c^2*d^4 + 72*c^4*d^2))/f^4))/4 + (243*c*d^14*(27*c^6 + 35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6)*((c*1i + d)/f
^3)^(1/3))/2 + (243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7)*((3^(1/2)
*1i)/2 + 1/2)*((c*1i + d)/(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)*((((3^(1/2)*1i)/2 + 1/2)*((34992*d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f
- 69984*c*d^14*((3^(1/2)*1i)/2 - 1/2)*(3*c^4 + 2*d^4 + 5*c^2*d^2)*(d^3/(c^2*f^3))^(1/3))*(d^3/(c^2*f^3))^(2/3)
)/9 - (1944*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f^3)*(d^3/(c^2*f^3))^(1/3))/3 + (486*d^14*(c + d*tan(e + f
*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 72*c^4*d^2))/f^4)*(d^3/(c^2*f^3))^(2/3))/9 - (243*c*d^14*(27*c^6 +
35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6)*(d^3/(c^2*f^3))^(1/3))/3 - (243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^
6 + 11*d^6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 - 1/2)*(d^3/(27*c^2*f^3))^(1/3) - log((((3^(1/2)*1
i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2)*((34992*d^15*(c + d*tan
(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f + 69984*c*d^14*((3^(1/2)*1i)/2 + 1/2)*(3*c^4 + 2*d^4 + 5*c^2*
d^2)*(d^3/(c^2*f^3))^(1/3))*(d^3/(c^2*f^3))^(2/3))/9 + (1944*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f^3)*(d^3
/(c^2*f^3))^(1/3))/3 + (486*d^14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 72*c^4*d^2))/f^4)
*(d^3/(c^2*f^3))^(2/3))/9 + (243*c*d^14*(27*c^6 + 35*d^6 + 89*c^2*d^4 + 81*c^4*d^2))/f^6)*(d^3/(c^2*f^3))^(1/3
))/3 + (243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^6 + 27*c^2*d^4 + 25*c^4*d^2))/f^7)*((3^(1/2)*1i)/2 +
1/2)*(d^3/(27*c^2*f^3))^(1/3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \cot ^{2}{\left (e + f x \right )}\, dx \]

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)

________________________________________________________________________________________