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

Optimal. Leaf size=373 \[ -\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{\sqrt{3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{2 f}+\frac{\sqrt{3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac{3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt [3]{c-i d}+\frac{1}{4} i x \sqrt [3]{c+i d} \]

[Out]

(-I/4)*(c - I*d)^(1/3)*x + (I/4)*(c + I*d)^(1/3)*x + (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) - (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) - (3*(c + d*Tan[e + f*x])^(1/3))/f - (9*c*(c + d*
Tan[e + f*x])^(4/3))/(28*d^2*f) + (3*Tan[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(7*d*f)

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Rubi [A]  time = 0.557804, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3566, 3630, 12, 3528, 3539, 3537, 57, 617, 204, 31} \[ -\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{\sqrt{3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{2 f}+\frac{\sqrt{3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac{3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt [3]{c-i d}+\frac{1}{4} i x \sqrt [3]{c+i d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-I/4)*(c - I*d)^(1/3)*x + (I/4)*(c + I*d)^(1/3)*x + (Sqrt[3]*(c - 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) - (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) - (3*(c + d*Tan[e + f*x])^(1/3))/f - (9*c*(c + d*
Tan[e + f*x])^(4/3))/(28*d^2*f) + (3*Tan[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(7*d*f)

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 3630

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3528

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

Rule 3539

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

Rule 3537

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 31

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

Rubi steps

\begin{align*} \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx &=\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}+\frac{3 \int \sqrt [3]{c+d \tan (e+f x)} \left (-c-\frac{7}{3} d \tan (e+f x)-c \tan ^2(e+f x)\right ) \, dx}{7 d}\\ &=-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}+\frac{3 \int -\frac{7}{3} d \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx}{7 d}\\ &=-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\\ &=-\frac{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\int \frac{-d+c \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx\\ &=-\frac{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac{1}{2} (-i c-d) \int \frac{1+i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx-\frac{1}{2} (i c-d) \int \frac{1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx\\ &=-\frac{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac{(c-i d) \operatorname{Subst}\left (\int \frac{1}{(-1+x) (c-i d x)^{2/3}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac{(c+i d) \operatorname{Subst}\left (\int \frac{1}{(-1+x) (c+i d x)^{2/3}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=-\frac{1}{4} i \sqrt [3]{c-i d} x+\frac{1}{4} i \sqrt [3]{c+i d} x-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}+\frac{\left (3 \sqrt [3]{c-i d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{c-i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac{\left (3 (c-i d)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(c-i d)^{2/3}+\sqrt [3]{c-i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac{\left (3 \sqrt [3]{c+i d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{c+i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac{\left (3 (c+i d)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(c+i d)^{2/3}+\sqrt [3]{c+i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}\\ &=-\frac{1}{4} i \sqrt [3]{c-i d} x+\frac{1}{4} i \sqrt [3]{c+i d} x-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{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{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+d \tan (e+f x)}}{f}-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac{\left (3 \sqrt [3]{c-i d}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}\right )}{2 f}-\frac{\left (3 \sqrt [3]{c+i d}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}\right )}{2 f}\\ &=-\frac{1}{4} i \sqrt [3]{c-i d} x+\frac{1}{4} i \sqrt [3]{c+i d} x+\frac{\sqrt{3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{2 f}+\frac{\sqrt{3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )}{2 f}-\frac{\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{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{3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}\\ \end{align*}

Mathematica [A]  time = 1.0805, size = 442, normalized size = 1.18 \[ \frac{-9 c^2 \sqrt [3]{c+d \tan (e+f x)}+14 \sqrt{3} d^2 \sqrt [3]{c-i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )+14 \sqrt{3} d^2 \sqrt [3]{c+i d} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt{3}}\right )+12 d^2 \tan ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)}-84 d^2 \sqrt [3]{c+d \tan (e+f x)}-14 d^2 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )-14 d^2 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )+7 d^2 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )+7 d^2 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )+3 c d \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{28 d^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*Sqrt[3]*(c - I*d)^(1/3)*d^2*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + 14*Sqrt
[3]*(c + I*d)^(1/3)*d^2*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 14*(c - I*d)^(1
/3)*d^2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 14*(c + I*d)^(1/3)*d^2*Log[(c + I*d)^(1/3) - (c +
d*Tan[e + f*x])^(1/3)] + 7*(c - I*d)^(1/3)*d^2*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)] + 7*(c + I*d)^(1/3)*d^2*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)] - 9*c^2*(c + d*Tan[e + f*x])^(1/3) - 84*d^2*(c + d*Tan[e + f*x])^(1/3
) + 3*c*d*Tan[e + f*x]*(c + d*Tan[e + f*x])^(1/3) + 12*d^2*Tan[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3))/(28*d^2*
f)

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Maple [C]  time = 0.026, size = 131, normalized size = 0.4 \begin{align*}{\frac{3}{7\,f{d}^{2}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{3}}}}-{\frac{3\,c}{4\,f{d}^{2}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{4}{3}}}}-3\,{\frac{\sqrt [3]{c+d\tan \left ( fx+e \right ) }}{f}}-{\frac{1}{2\,f}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-2\,{{\it \_Z}}^{3}c+{c}^{2}+{d}^{2} \right ) }{\frac{{{\it \_R}}^{3}c-{c}^{2}-{d}^{2}}{{{\it \_R}}^{5}-{{\it \_R}}^{2}c}\ln \left ( \sqrt [3]{c+d\tan \left ( fx+e \right ) }-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7/f/d^2*(c+d*tan(f*x+e))^(7/3)-3/4*c*(c+d*tan(f*x+e))^(4/3)/d^2/f-3*(c+d*tan(f*x+e))^(1/3)/f-1/2/f*sum((_R^3
*c-c^2-d^2)/(_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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^3*(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)^3, x)

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Fricas [B]  time = 2.56462, size = 7194, normalized size = 19.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(14*d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(
d^2/f^6)/d^2))*log(2*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*
arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e
) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) - 56*d^2*f*((c^2 + d^2)/f^6)^(1/6)*arctan((sqrt(2*f*((c*cos(f*x + e)
+ d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^
3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3))
*f^5*((c^2 + d^2)/f^6)^(5/6) - f^5*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5
/6) - (c^2 + d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))/((c^2 + d^2)*sin(2/3
*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2))))*cos(f*x + e)^2*sin(2/3*arctan((f^6*sqrt((c^2
+ d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - 28*(sqrt(3)*d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*cos(2/3*a
rctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*s
in(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*arctan(-(2*sqrt(3)*f^5*((c*cos(f*x + e)
+ d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f
^3)*sqrt(d^2/f^6)/d^2)) + 2*(f^5*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6
) - 2*(c^2 + d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*sin(2/3*arctan((f^6*
sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - 2*(sqrt(3)*f^5*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^
6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^5*((c^2 + d^2)/f^6)^(5/6)*sin(2/3*arctan((f^6*sqrt((c
^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*sqrt(-sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^
(1/3)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*c
os(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)
/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x
+ e))^(2/3)) - sqrt(3)*(c^2 + d^2))/(4*(c^2 + d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2
/f^6)/d^2))^2 - c^2 - d^2)) + 28*(sqrt(3)*d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*cos(2/3*arctan((f^6*sqr
t((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*sin(2/3*arctan(
(f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*arctan((2*sqrt(3)*f^5*((c*cos(f*x + e) + d*sin(f*x +
e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^
6)/d^2)) - 2*(f^5*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6) - 2*(c^2 + d^
2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*sin(2/3*arctan((f^6*sqrt((c^2 + d^2
)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - 2*(sqrt(3)*f^5*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d
^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f^5*((c^2 + d^2)/f^6)^(5/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6)
+ c*f^3)*sqrt(d^2/f^6)/d^2)))*sqrt(sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^
2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*cos(f*x + e) + d*
sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*s
qrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) - s
qrt(3)*(c^2 + d^2))/(4*(c^2 + d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2))^2 -
c^2 - d^2)) + 7*(sqrt(3)*d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6
) + c*f^3)*sqrt(d^2/f^6)/d^2)) - d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*cos(2/3*arctan((f^6*sqrt((c^2 +
d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*log(sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*(
(c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*cos(f*x
+ e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) +
c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(
2/3)) - 7*(sqrt(3)*d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*
f^3)*sqrt(d^2/f^6)/d^2)) + d^2*f*((c^2 + d^2)/f^6)^(1/6)*cos(f*x + e)^2*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f
^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*log(-sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2
+ d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*cos(f*x + e)
+ d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^
3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3))
+ 3*(c*d*cos(f*x + e)*sin(f*x + e) - (3*c^2 + 32*d^2)*cos(f*x + e)^2 + 4*d^2)*((c*cos(f*x + e) + d*sin(f*x +
e))/cos(f*x + e))^(1/3))/(d^2*f*cos(f*x + e)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 3.08886, size = 589, normalized size = 1.58 \begin{align*} -\frac{1}{4} \,{\left (i \, \sqrt{3} + 1\right )} \left (-\frac{c - i \, d}{f^{3}}\right )^{\frac{1}{3}} \log \left (-{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{16} f^{7}{\left (\sqrt{3} + i\right )} + 2 \,{\left (i \, c + d\right )}^{\frac{1}{3}} d^{16} f^{7}\right ) - \frac{1}{4} \,{\left (i \, \sqrt{3} + 1\right )} \left (-\frac{c + i \, d}{f^{3}}\right )^{\frac{1}{3}} \log \left (-{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{16} f^{7}{\left (\sqrt{3} + i\right )} + 2 \,{\left (i \, c - d\right )}^{\frac{1}{3}} d^{16} f^{7}\right ) - \frac{1}{4} \,{\left (-i \, \sqrt{3} + 1\right )} \left (-\frac{c - i \, d}{f^{3}}\right )^{\frac{1}{3}} \log \left ({\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{16} f^{7}{\left (\sqrt{3} - i\right )} + 2 \,{\left (i \, c + d\right )}^{\frac{1}{3}} d^{16} f^{7}\right ) - \frac{1}{4} \,{\left (-i \, \sqrt{3} + 1\right )} \left (-\frac{c + i \, d}{f^{3}}\right )^{\frac{1}{3}} \log \left ({\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{16} f^{7}{\left (\sqrt{3} - i\right )} + 2 \,{\left (i \, c - d\right )}^{\frac{1}{3}} d^{16} f^{7}\right ) + \frac{1}{2} \, \left (-\frac{c - i \, d}{f^{3}}\right )^{\frac{1}{3}} \log \left (i \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{16} f^{7} +{\left (i \, c + d\right )}^{\frac{1}{3}} d^{16} f^{7}\right ) + \frac{1}{2} \, \left (-\frac{c + i \, d}{f^{3}}\right )^{\frac{1}{3}} \log \left (i \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{16} f^{7} +{\left (i \, c - d\right )}^{\frac{1}{3}} d^{16} f^{7}\right ) + \frac{3 \,{\left (4 \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{7}{3}} d^{12} f^{6} - 7 \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{4}{3}} c d^{12} f^{6} - 28 \,{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}} d^{14} f^{6}\right )}}{28 \, d^{14} f^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(I*sqrt(3) + 1)*(-(c - I*d)/f^3)^(1/3)*log(-(d*tan(f*x + e) + c)^(1/3)*d^16*f^7*(sqrt(3) + I) + 2*(I*c +
d)^(1/3)*d^16*f^7) - 1/4*(I*sqrt(3) + 1)*(-(c + I*d)/f^3)^(1/3)*log(-(d*tan(f*x + e) + c)^(1/3)*d^16*f^7*(sqrt
(3) + I) + 2*(I*c - d)^(1/3)*d^16*f^7) - 1/4*(-I*sqrt(3) + 1)*(-(c - I*d)/f^3)^(1/3)*log((d*tan(f*x + e) + c)^
(1/3)*d^16*f^7*(sqrt(3) - I) + 2*(I*c + d)^(1/3)*d^16*f^7) - 1/4*(-I*sqrt(3) + 1)*(-(c + I*d)/f^3)^(1/3)*log((
d*tan(f*x + e) + c)^(1/3)*d^16*f^7*(sqrt(3) - I) + 2*(I*c - d)^(1/3)*d^16*f^7) + 1/2*(-(c - I*d)/f^3)^(1/3)*lo
g(I*(d*tan(f*x + e) + c)^(1/3)*d^16*f^7 + (I*c + d)^(1/3)*d^16*f^7) + 1/2*(-(c + I*d)/f^3)^(1/3)*log(I*(d*tan(
f*x + e) + c)^(1/3)*d^16*f^7 + (I*c - d)^(1/3)*d^16*f^7) + 3/28*(4*(d*tan(f*x + e) + c)^(7/3)*d^12*f^6 - 7*(d*
tan(f*x + e) + c)^(4/3)*c*d^12*f^6 - 28*(d*tan(f*x + e) + c)^(1/3)*d^14*f^6)/(d^14*f^7)