3.234 \(\int \frac{\tan ^{\frac{8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx\)

Optimal. Leaf size=319 \[ -\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{5 \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{12 a d}-\frac{2 i \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{\sqrt{3} a d}+\frac{5 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{2 i \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d}+\frac{5 \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{5 \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{i \log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d} \]

[Out]

(-5*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d) + (5*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (
(2*I)*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(Sqrt[3]*a*d) + (5*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) + (((
2*I)/3)*Log[1 + Tan[c + d*x]^(2/3)])/(a*d) + (5*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*S
qrt[3]*a*d) - (5*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - ((I/3)*Log[1 - Ta
n[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(a*d) - ((2*I)*Tan[c + d*x]^(2/3))/(a*d) - Tan[c + d*x]^(5/3)/(2*d*(a
+ I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.495587, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3550, 3528, 3538, 3476, 329, 275, 200, 31, 634, 618, 204, 628, 295, 203} \[ -\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{5 \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{12 a d}-\frac{2 i \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{\sqrt{3} a d}+\frac{5 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{2 i \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d}+\frac{5 \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{5 \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{i \log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^(8/3)/(a + I*a*Tan[c + d*x]),x]

[Out]

(-5*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d) + (5*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (
(2*I)*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(Sqrt[3]*a*d) + (5*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) + (((
2*I)/3)*Log[1 + Tan[c + d*x]^(2/3)])/(a*d) + (5*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*S
qrt[3]*a*d) - (5*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - ((I/3)*Log[1 - Ta
n[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(a*d) - ((2*I)*Tan[c + d*x]^(2/3))/(a*d) - Tan[c + d*x]^(5/3)/(2*d*(a
+ I*a*Tan[c + d*x]))

Rule 3550

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

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 3538

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

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\tan ^{\frac{8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{\int \tan ^{\frac{2}{3}}(c+d x) \left (\frac{5 a}{3}-\frac{8}{3} i a \tan (c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{\int \frac{\frac{8 i a}{3}+\frac{5}{3} a \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{(4 i) \int \frac{1}{\sqrt [3]{\tan (c+d x)}} \, dx}{3 a}+\frac{5 \int \tan ^{\frac{2}{3}}(c+d x) \, dx}{6 a}\\ &=-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{6 a d}\\ &=-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{a d}+\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 a d}\\ &=-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{1+x^3} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{a d}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}\\ &=\frac{5 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{3 a d}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{2-x}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{3 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{24 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{24 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{8 \sqrt{3} a d}-\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{8 \sqrt{3} a d}\\ &=\frac{5 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{2 i \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{3 a d}+\frac{5 \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{5 \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{i \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{3 a d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{a d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}\\ &=-\frac{5 \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{2 i \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{3 a d}+\frac{5 \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{5 \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{i \log \left (1-\tan ^{\frac{2}{3}}(c+d x)+\tan ^{\frac{4}{3}}(c+d x)\right )}{3 a d}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac{2}{3}}(c+d x)\right )}{a d}\\ &=-\frac{5 \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac{2 i \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{\sqrt{3} a d}+\frac{5 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{2 i \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{3 a d}+\frac{5 \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{5 \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{i \log \left (1-\tan ^{\frac{2}{3}}(c+d x)+\tan ^{\frac{4}{3}}(c+d x)\right )}{3 a d}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 1.08834, size = 163, normalized size = 0.51 \[ \frac{i e^{-2 i (c+d x)} \left (3 \sqrt [3]{2} e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left (1-e^{2 i (c+d x)}\right )\right )+26 e^{2 i (c+d x)} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )-28 e^{2 i (c+d x)}-4\right ) \tan ^{\frac{2}{3}}(c+d x)}{16 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^(8/3)/(a + I*a*Tan[c + d*x]),x]

[Out]

((I/16)*(-4 - 28*E^((2*I)*(c + d*x)) + 3*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeom
etric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 26*E^((2*I)*(c + d*x))*Hypergeometric2F1[2/3, 1, 5/3, -
((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))])*Tan[c + d*x]^(2/3))/(a*d*E^((2*I)*(c + d*x)))

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Maple [A]  time = 0.043, size = 275, normalized size = 0.9 \begin{align*}{\frac{-{\frac{3\,i}{2}}}{ad} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}}-{\frac{1}{3\,ad}\sqrt [3]{\tan \left ( dx+c \right ) } \left ( -i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) ^{-1}}+{\frac{{\frac{i}{6}}}{ad} \left ( -i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) ^{-1}}-{\frac{{\frac{13\,i}{24}}}{ad}\ln \left ( -i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) }-{\frac{13\,\sqrt{3}}{12\,ad}{\it Artanh} \left ({\frac{\sqrt{3}}{3} \left ( -i+2\,\sqrt [3]{\tan \left ( dx+c \right ) } \right ) } \right ) }-{\frac{{\frac{i}{8}}}{ad}\ln \left ( i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) }+{\frac{\sqrt{3}}{4\,ad}{\it Artanh} \left ({\frac{\sqrt{3}}{3} \left ( i+2\,\sqrt [3]{\tan \left ( dx+c \right ) } \right ) } \right ) }+{\frac{{\frac{i}{4}}}{ad}\ln \left ( \sqrt [3]{\tan \left ( dx+c \right ) }-i \right ) }+{\frac{{\frac{13\,i}{12}}}{ad}\ln \left ( \sqrt [3]{\tan \left ( dx+c \right ) }+i \right ) }-{\frac{1}{6\,ad} \left ( \sqrt [3]{\tan \left ( dx+c \right ) }+i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(8/3)/(a+I*a*tan(d*x+c)),x)

[Out]

-3/2*I/d/a*tan(d*x+c)^(2/3)-1/3/d/a/(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)*tan(d*x+c)^(1/3)+1/6*I/d/a/(-I*ta
n(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-13/24*I/d/a*ln(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-13/12/d/a*3^(1/2)*ar
ctanh(1/3*(-I+2*tan(d*x+c)^(1/3))*3^(1/2))-1/8*I/d/a*ln(I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)+1/4/d/a*3^(1/2)
*arctanh(1/3*(I+2*tan(d*x+c)^(1/3))*3^(1/2))+1/4*I/d/a*ln(tan(d*x+c)^(1/3)-I)+13/12*I/d/a*ln(tan(d*x+c)^(1/3)+
I)-1/6/d/a/(tan(d*x+c)^(1/3)+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(8/3)/(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.7643, size = 1511, normalized size = 4.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(8/3)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/24*((3*sqrt(3)*a*d*sqrt(1/(a^2*d^2))*e^(2*I*d*x + 2*I*c) - 3*I*e^(2*I*d*x + 2*I*c))*log(1/2*sqrt(3)*a*d*sqrt
(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) + 1/2*I) - (3*sqrt(3)*a*d*sqrt(
1/(a^2*d^2))*e^(2*I*d*x + 2*I*c) + 3*I*e^(2*I*d*x + 2*I*c))*log(-1/2*sqrt(3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2
*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) + 1/2*I) - (39*sqrt(1/3)*a*d*sqrt(1/(a^2*d^2))*e^(2*I*d*
x + 2*I*c) + 13*I*e^(2*I*d*x + 2*I*c))*log(3/2*sqrt(1/3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)
/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - 1/2*I) + (39*sqrt(1/3)*a*d*sqrt(1/(a^2*d^2))*e^(2*I*d*x + 2*I*c) - 13*I*e^
(2*I*d*x + 2*I*c))*log(-3/2*sqrt(1/3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*
c) + 1))^(1/3) - 1/2*I) + 26*I*e^(2*I*d*x + 2*I*c)*log(((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)
)^(1/3) + I) + 6*I*e^(2*I*d*x + 2*I*c)*log(((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - I)
+ ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(-42*I*e^(2*I*d*x + 2*I*c) - 6*I))*e^(-2*I*d
*x - 2*I*c)/(a*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(8/3)/(a+I*a*tan(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.21022, size = 312, normalized size = 0.98 \begin{align*} \frac{13 \, \sqrt{3} \log \left (-\frac{\sqrt{3} - 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} + i}{\sqrt{3} + 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} - i}\right )}{24 \, a d} - \frac{\sqrt{3} \log \left (-\frac{\sqrt{3} - 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} - i}{\sqrt{3} + 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} + i}\right )}{8 \, a d} - \frac{i \, \log \left (\tan \left (d x + c\right )^{\frac{2}{3}} + i \, \tan \left (d x + c\right )^{\frac{1}{3}} - 1\right )}{8 \, a d} - \frac{13 i \, \log \left (\tan \left (d x + c\right )^{\frac{2}{3}} - i \, \tan \left (d x + c\right )^{\frac{1}{3}} - 1\right )}{24 \, a d} + \frac{13 i \, \log \left (\tan \left (d x + c\right )^{\frac{1}{3}} + i\right )}{12 \, a d} + \frac{i \, \log \left (\tan \left (d x + c\right )^{\frac{1}{3}} - i\right )}{4 \, a d} - \frac{3 i \, \tan \left (d x + c\right )^{\frac{2}{3}}}{2 \, a d} - \frac{\tan \left (d x + c\right )^{\frac{2}{3}}}{2 \, a d{\left (\tan \left (d x + c\right ) - i\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(8/3)/(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

13/24*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) + I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) - I))/(a*d) - 1/8*sqrt
(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) - I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) + I))/(a*d) - 1/8*I*log(tan(d*x
+ c)^(2/3) + I*tan(d*x + c)^(1/3) - 1)/(a*d) - 13/24*I*log(tan(d*x + c)^(2/3) - I*tan(d*x + c)^(1/3) - 1)/(a*d
) + 13/12*I*log(tan(d*x + c)^(1/3) + I)/(a*d) + 1/4*I*log(tan(d*x + c)^(1/3) - I)/(a*d) - 3/2*I*tan(d*x + c)^(
2/3)/(a*d) - 1/2*tan(d*x + c)^(2/3)/(a*d*(tan(d*x + c) - I))