3.681 \(\int \tan ^4(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\)

Optimal. Leaf size=525 \[ \frac {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {\sqrt {3} \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 d f}+\frac {\sqrt {3} \sqrt {-d^2} \sqrt [3]{c+\sqrt {-d^2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 d f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d f}-\frac {3 \sqrt {-d^2} \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d f}+\frac {\sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {\sqrt {-d^2} \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {1}{4} x \sqrt [3]{c-\sqrt {-d^2}}-\frac {1}{4} x \sqrt [3]{c+\sqrt {-d^2}}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f} \]

[Out]

-1/4*x*(c-(-d^2)^(1/2))^(1/3)+1/4*ln(cos(f*x+e))*(c-(-d^2)^(1/2))^(1/3)*(-d^2)^(1/2)/d/f+3/4*ln((c-(-d^2)^(1/2
))^(1/3)-(c+d*tan(f*x+e))^(1/3))*(c-(-d^2)^(1/2))^(1/3)*(-d^2)^(1/2)/d/f-1/2*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)*(-d^2)^(1/2)/d/f-1/4*x*(c+(-d^2)^(1/2))^
(1/3)-1/4*ln(cos(f*x+e))*(-d^2)^(1/2)*(c+(-d^2)^(1/2))^(1/3)/d/f-3/4*ln((c+(-d^2)^(1/2))^(1/3)-(c+d*tan(f*x+e)
)^(1/3))*(-d^2)^(1/2)*(c+(-d^2)^(1/2))^(1/3)/d/f+1/2*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)*(-d^2)^(1/2)*(c+(-d^2)^(1/2))^(1/3)/d/f+3/140*(9*c^2-35*d^2)*(c+d*tan(f*x+e))^(4/3)/d^3
/f-9/35*c*tan(f*x+e)*(c+d*tan(f*x+e))^(4/3)/d^2/f+3/10*tan(f*x+e)^2*(c+d*tan(f*x+e))^(4/3)/d/f

________________________________________________________________________________________

Rubi [A]  time = 0.78, antiderivative size = 525, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3566, 3647, 3631, 3485, 712, 50, 57, 617, 204, 31} \[ \frac {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {\sqrt {3} \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 d f}+\frac {\sqrt {3} \sqrt {-d^2} \sqrt [3]{c+\sqrt {-d^2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 d f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d f}-\frac {3 \sqrt {-d^2} \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d f}+\frac {\sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {\sqrt {-d^2} \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {1}{4} x \sqrt [3]{c-\sqrt {-d^2}}-\frac {1}{4} x \sqrt [3]{c+\sqrt {-d^2}}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f} \]

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^4*(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 - (Sqrt[3]*Sqrt[-d^2]*(c - Sqrt[-d^2])^(1/3)*ArcT
an[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*d*f) + (Sqrt[3]*Sqrt[-d^2]*(c + Sq
rt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*d*f) + (Sqrt[-
d^2]*(c - Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*d*f) - (Sqrt[-d^2]*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])
/(4*d*f) + (3*Sqrt[-d^2]*(c - Sqrt[-d^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*d
*f) - (3*Sqrt[-d^2]*(c + Sqrt[-d^2])^(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*d*f) +
(3*(9*c^2 - 35*d^2)*(c + d*Tan[e + f*x])^(4/3))/(140*d^3*f) - (9*c*Tan[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(
35*d^2*f) + (3*Tan[e + f*x]^2*(c + d*Tan[e + f*x])^(4/3))/(10*d*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 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3631

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

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^m*(c + d
*Tan[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, 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[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps

\begin {align*} \int \tan ^4(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx &=\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f}+\frac {3 \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \left (-2 c-\frac {10}{3} d \tan (e+f x)-2 c \tan ^2(e+f x)\right ) \, dx}{10 d}\\ &=-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f}+\frac {9 \int \sqrt [3]{c+d \tan (e+f x)} \left (2 c^2+\frac {2}{9} \left (9 c^2-35 d^2\right ) \tan ^2(e+f x)\right ) \, dx}{70 d^2}\\ &=\frac {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f}+\int \sqrt [3]{c+d \tan (e+f x)} \, dx\\ &=\frac {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d 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 {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d 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 {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d 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 {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d 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 {\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 {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d 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 {\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 {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 {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d 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 {\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 {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 {3 \left (9 c^2-35 d^2\right ) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac {9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac {3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 11.38, size = 371, normalized size = 0.71 \[ \frac {\frac {3 \sqrt [3]{c+d \tan (e+f x)} \left (9 c^3-d \left (3 c^2+49 d^2\right ) \tan (e+f x)+2 d^2 \sec ^2(e+f x) (c+7 d \tan (e+f x))-37 c d^2\right )}{35 d^3}-i \sqrt [3]{c-i d} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )+\log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )\right )+i \sqrt [3]{c+i d} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )+\log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )\right )}{4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^4*(c + d*Tan[e + f*x])^(1/3),x]

[Out]

((-I)*(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*(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)]) + (3*(c + d*Tan[e + f*x])^(1/3)*(9*c^3 - 37*
c*d^2 - d*(3*c^2 + 49*d^2)*Tan[e + f*x] + 2*d^2*Sec[e + f*x]^2*(c + 7*d*Tan[e + f*x])))/(35*d^3))/(4*f)

________________________________________________________________________________________

fricas [B]  time = 1.12, size = 3557, normalized size = 6.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4*(c+d*tan(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

1/140*(70*d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^3*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6)
+ d*f^3*sqrt(c^2/f^6))/c^2))*log(2*c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)
*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2))
+ c^2*f^2*((c^2 + d^2)/f^6)^(1/3) + c^2*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) + 280*d^3*f*((
c^2 + d^2)/f^6)^(1/6)*arctan(-(c*f^8*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^
2 + d^2)/f^6)^(5/6) - sqrt(2*c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2
+ d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + c^2*f
^2*((c^2 + d^2)/f^6)^(1/3) + c^2*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3))*f^8*sqrt(c^2/f^6)*((c
^2 + d^2)/f^6)^(5/6) + (c^4 + c^2*d^2)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^
2/f^6))/c^2)))/((c^4 + c^2*d^2)*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))
/c^2))))*cos(f*x + e)^3*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) +
140*(sqrt(3)*d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^3*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/
f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^3*sin(2/3*arctan((f^6*sqrt(c^2/
f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)))*arctan(-(2*c*f^8*((c*cos(f*x + e) + d*sin(f*x + e))/c
os(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^
6) + d*f^3*sqrt(c^2/f^6))/c^2)) - 2*(sqrt(3)*c*f^8*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt
(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6) + 2*(c^4 + c^2*d^2)*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6)
+ d*f^3*sqrt(c^2/f^6))/c^2)))*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c
^2)) + 2*(sqrt(3)*f^8*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)
/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - f^8*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt(c^2/f^
6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)))*sqrt(sqrt(3)*c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/c
os(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^
6) + d*f^3*sqrt(c^2/f^6))/c^2)) - c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*(
(c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) +
c^2*f^2*((c^2 + d^2)/f^6)^(1/3) + c^2*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) + sqrt(3)*(c^4 +
c^2*d^2))/(3*c^4 + 3*c^2*d^2 - 4*(c^4 + c^2*d^2)*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*
f^3*sqrt(c^2/f^6))/c^2))^2)) + 140*(sqrt(3)*d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^3*cos(2/3*arctan((f^6*s
qrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^3
*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)))*arctan((2*c*f^8*((c*cos
(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt
(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + 2*(sqrt(3)*c*f^8*((c*cos(f*x + e) + d*sin(f*x +
e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6) - 2*(c^4 + c^2*d^2)*cos(2/3*arctan((f^6*sqrt(c^
2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)))*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/
f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - 2*(sqrt(3)*f^8*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6)*sin(2/3*arctan((f^6*s
qrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + f^8*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(5/6)*co
s(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)))*sqrt(-sqrt(3)*c*f^4*((c*co
s(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqr
t(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f
*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) +
d*f^3*sqrt(c^2/f^6))/c^2)) + c^2*f^2*((c^2 + d^2)/f^6)^(1/3) + c^2*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x
+ e))^(2/3)) - sqrt(3)*(c^4 + c^2*d^2))/(3*c^4 + 3*c^2*d^2 - 4*(c^4 + c^2*d^2)*cos(2/3*arctan((f^6*sqrt(c^2/f
^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2))^2)) - 35*(sqrt(3)*d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x
+ e)^3*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + d^3*f*((c^2 + d
^2)/f^6)^(1/6)*cos(f*x + e)^3*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c
^2)))*log(sqrt(3)*c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)
^(1/6)*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - c*f^4*((c*cos(f*
x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt(c^
2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + c^2*f^2*((c^2 + d^2)/f^6)^(1/3) + c^2*((c*cos(f*x
+ e) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) + 35*(sqrt(3)*d^3*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^3*sin(2/3
*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - d^3*f*((c^2 + d^2)/f^6)^(1/6)*
cos(f*x + e)^3*cos(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)))*log(-sqrt
(3)*c*f^4*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3
*arctan((f^6*sqrt(c^2/f^6)*sqrt((c^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) - c*f^4*((c*cos(f*x + e) + d*sin
(f*x + e))/cos(f*x + e))^(1/3)*sqrt(c^2/f^6)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt(c^2/f^6)*sqrt((c
^2 + d^2)/f^6) + d*f^3*sqrt(c^2/f^6))/c^2)) + c^2*f^2*((c^2 + d^2)/f^6)^(1/3) + c^2*((c*cos(f*x + e) + d*sin(f
*x + e))/cos(f*x + e))^(2/3)) + 3*(2*c*d^2*cos(f*x + e) + (9*c^3 - 37*c*d^2)*cos(f*x + e)^3 + (14*d^3 - (3*c^2
*d + 49*d^3)*cos(f*x + e)^2)*sin(f*x + e))*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3))/(d^3*f*cos(
f*x + e)^3)

________________________________________________________________________________________

giac [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(tan(f*x+e)^4*(c+d*tan(f*x+e))^(1/3),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [C]  time = 0.66, size = 145, normalized size = 0.28 \[ \frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {10}{3}}}{10 f \,d^{3}}-\frac {6 c \left (c +d \tan \left (f x +e \right )\right )^{\frac {7}{3}}}{7 f \,d^{3}}+\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {4}{3}} c^{2}}{4 f \,d^{3}}-\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}{4 d f}+\frac {d \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2 f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^4*(c+d*tan(f*x+e))^(1/3),x)

[Out]

3/10/f/d^3*(c+d*tan(f*x+e))^(10/3)-6/7/f/d^3*c*(c+d*tan(f*x+e))^(7/3)+3/4/f/d^3*(c+d*tan(f*x+e))^(4/3)*c^2-3/4
*(c+d*tan(f*x+e))^(4/3)/d/f+1/2/f*d*sum(_R^3/(_R^5-_R^2*c)*ln((c+d*tan(f*x+e))^(1/3)-_R),_R=RootOf(_Z^6-2*_Z^3
*c+c^2+d^2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4*(c+d*tan(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

integrate((d*tan(f*x + e) + c)^(1/3)*tan(f*x + e)^4, x)

________________________________________________________________________________________

mupad [B]  time = 18.88, size = 1015, normalized size = 1.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^4*(c + d*tan(e + f*x))^(1/3),x)

[Out]

log((c + d*tan(e + f*x))^(1/3) + f*(-(c*1i + d)/f^3)^(1/3)*1i)*(-(c*1i + d)/(8*f^3))^(1/3) + (c + d*tan(e + f*
x))^(1/3)*(2*c*((6*c^2)/(d^3*f) - (3*(c^2 + d^2))/(d^3*f)) - (12*c^3)/(d^3*f) + (6*c*(c^2 + d^2))/(d^3*f)) + (
(3*c^2)/(2*d^3*f) - (3*(c^2 + d^2))/(4*d^3*f))*(c + d*tan(e + f*x))^(4/3) + log(d*(c + d*tan(e + f*x))^(1/3)*1
i - c*(c + d*tan(e + f*x))^(1/3) + f^4*((c*1i - d)/f^3)^(4/3) + 2*d*f*((c*1i - d)/f^3)^(1/3))*((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
)*((3888*d^5*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f - 3888*c*d^4*((c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/2 + 1
/2)*(c^2 + d^2)))/4 + (1944*c*d^5*(c^2 + d^2))/f^3))/2 - (486*(d^8 - c^4*d^4)*(c + d*tan(e + f*x))^(1/3))/f^4)
*((3^(1/2)*1i)/2 + 1/2)*((c*1i - d)/(8*f^3))^(1/3) + log((486*(d^8 - c^4*d^4)*(c + d*tan(e + f*x))^(1/3))/f^4
- (((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)*((3888*d^5*(c
^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f + 3888*c*d^4*((c*1i - d)/f^3)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(c^2 + d^2)
))/4 - (1944*c*d^5*(c^2 + d^2))/f^3))/2)*((3^(1/2)*1i)/2 - 1/2)*((c*1i - d)/(8*f^3))^(1/3) + (3*(c + d*tan(e +
f*x))^(10/3))/(10*d^3*f) - log((((3^(1/2)*1i)/2 + 1/2)*((((3888*d^5*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f
- 3888*c*d^4*((3^(1/2)*1i)/2 + 1/2)*(-(c*1i + d)/f^3)^(1/3)*(c^2 + d^2))*((3^(1/2)*1i)/2 - 1/2)*(-(c*1i + d)/
f^3)^(2/3))/4 + (1944*c*d^5*(c^2 + d^2))/f^3)*(-(c*1i + d)/f^3)^(1/3))/2 - (486*(d^8 - c^4*d^4)*(c + d*tan(e +
f*x))^(1/3))/f^4)*((3^(1/2)*1i)/2 + 1/2)*(-(c*1i + d)/(8*f^3))^(1/3) + log((486*(d^8 - c^4*d^4)*(c + d*tan(e
+ f*x))^(1/3))/f^4 - (((3^(1/2)*1i)/2 - 1/2)*((((3888*d^5*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f + 3888*c*d
^4*((3^(1/2)*1i)/2 - 1/2)*(-(c*1i + d)/f^3)^(1/3)*(c^2 + d^2))*((3^(1/2)*1i)/2 + 1/2)*(-(c*1i + d)/f^3)^(2/3))
/4 - (1944*c*d^5*(c^2 + d^2))/f^3)*(-(c*1i + d)/f^3)^(1/3))/2)*((3^(1/2)*1i)/2 - 1/2)*(-(c*1i + d)/(8*f^3))^(1
/3) - (6*c*(c + d*tan(e + f*x))^(7/3))/(7*d^3*f)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**4*(c+d*tan(f*x+e))**(1/3),x)

[Out]

Integral((c + d*tan(e + f*x))**(1/3)*tan(e + f*x)**4, x)

________________________________________________________________________________________