3.7.83 \(\int \tan ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\) [683]

Optimal. Leaf size=439 \[ \frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {\sqrt {3} d \sqrt [3]{c-\sqrt {-d^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {\sqrt {3} d \sqrt [3]{c+\sqrt {-d^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {d \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}+\frac {3 d \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}+\frac {3 (c+d \tan (e+f x))^{4/3}}{4 d f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3624, 3566, 726, 52, 59, 631, 210, 31} \begin {gather*} -\frac {\sqrt {3} d \sqrt [3]{c-\sqrt {-d^2}} \text {ArcTan}\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}} \text {ArcTan}\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 {3 (c+d \tan (e+f x))^{4/3}}{4 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[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 - (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) + (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) +
(3*(c + d*Tan[e + f*x])^(4/3))/(4*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 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 726

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 3566

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 3624

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.82, size = 313, normalized size = 0.71 \begin {gather*} \frac {i \sqrt [3]{c-i d} \left (2 \sqrt {3} \text {ArcTan}\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-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\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 )\right )-i \sqrt [3]{c+i d} \left (2 \sqrt {3} \text {ArcTan}\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+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\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 )\right )+\frac {3 (c+d \tan (e+f x))^{4/3}}{d}}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^2*(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])^(4/3))/d)/(4*f)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.23, size = 82, normalized size = 0.19

method	result	size
derivativedivides	\(\frac {\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}{4}-\frac {d^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2}}{d f}\)	\(82\)
default	\(\frac {\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}{4}-\frac {d^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2}}{d f}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/f/d*(1/4*(c+d*tan(f*x+e))^(4/3)-1/6*d^2*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(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)*tan(f*x + e)^2, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3577 vs. \(2 (352) = 704\).
time = 1.29, size = 3577, normalized size = 8.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*d*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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)) + 8*d*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))))*co
s(f*x + e)*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)) + 4*(sqrt(3)*d
*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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*s
qrt(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))/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)) + 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)) + 4*(sqrt(3)*d*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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*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)*s
qrt((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))/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)) - 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)) - (sqrt(3)*d*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)*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)*sqr
t((c^2 + d^2)/f^6) - d*f^3*sqrt(c^2/f^6))/c^2))...

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

________________________________________________________________________________________

Mupad [B]
time = 8.73, size = 881, normalized size = 2.01 \begin {gather*} \ln \left ({\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}+f\,{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,{\left (-\frac {-d+c\,1{}\mathrm {i}}{8\,f^3}\right )}^{1/3}+\ln \left (c\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}+d\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}\,1{}\mathrm {i}-f^4\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{4/3}+2\,d\,f\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{8\,f^3}\right )}^{1/3}+\ln \left (-\frac {486\,\left (d^8-c^4\,d^4\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f^4}-\frac {{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3888\,d^5\,\left (c^2+d^2\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f}-3888\,c\,d^4\,{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (c^2+d^2\right )\right )}{4}-\frac {1944\,c\,d^5\,\left (c^2+d^2\right )}{f^3}\right )}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {-d+c\,1{}\mathrm {i}}{8\,f^3}\right )}^{1/3}-\ln \left (\frac {486\,\left (d^8-c^4\,d^4\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f^4}+\frac {{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3888\,d^5\,\left (c^2+d^2\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f}+3888\,c\,d^4\,{\left (-\frac {-d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (c^2+d^2\right )\right )}{4}+\frac {1944\,c\,d^5\,\left (c^2+d^2\right )}{f^3}\right )}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {-d+c\,1{}\mathrm {i}}{8\,f^3}\right )}^{1/3}+\frac {3\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{4/3}}{4\,d\,f}+\ln \left (-\frac {486\,\left (d^8-c^4\,d^4\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f^4}-\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\left (\frac {3888\,d^5\,\left (c^2+d^2\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f}-3888\,c\,d^4\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,\left (c^2+d^2\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{2/3}}{4}-\frac {1944\,c\,d^5\,\left (c^2+d^2\right )}{f^3}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{8\,f^3}\right )}^{1/3}-\ln \left (\frac {486\,\left (d^8-c^4\,d^4\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f^4}+\frac {\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\left (\frac {3888\,d^5\,\left (c^2+d^2\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{1/3}}{f}+3888\,c\,d^4\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}\,\left (c^2+d^2\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{2/3}}{4}+\frac {1944\,c\,d^5\,\left (c^2+d^2\right )}{f^3}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{f^3}\right )}^{1/3}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {d+c\,1{}\mathrm {i}}{8\,f^3}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(f*(-(c*1i - d)/f^3)^(1/3)*1i + (c + d*tan(e + f*x))^(1/3))*(-(c*1i - d)/(8*f^3))^(1/3) + log(c*(c + d*tan(
e + f*x))^(1/3) + d*(c + d*tan(e + f*x))^(1/3)*1i - 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(- (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*ta
n(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) - log((486*(d^8 - c^4*d^4)*(c + d*ta
n(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)*1
i)/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))^(4/3))/(4*d*f) + 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) - 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)

________________________________________________________________________________________