3.238 \(\int \frac{1}{\tan ^{\frac{5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx\)

Optimal. Leaf size=321 \[ \frac{5 i \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac{5 i \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{12 a d}+\frac{2 \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{\sqrt{3} a d}-\frac{5 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{2}{a d \tan ^{\frac{2}{3}}(c+d x)}+\frac{1}{2 d \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{2 \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d}+\frac{5 i \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{5 i \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{\log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.417109, antiderivative size = 321, 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 = {3552, 3529, 3538, 3476, 329, 209, 634, 618, 204, 628, 203, 275, 292, 31} \[ \frac{5 i \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac{5 i \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{12 a d}+\frac{2 \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{\sqrt{3} a d}-\frac{5 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{2}{a d \tan ^{\frac{2}{3}}(c+d x)}+\frac{1}{2 d \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{2 \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d}+\frac{5 i \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{5 i \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{\log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(5/3)*(a + I*a*Tan[c + d*x])),x]

[Out]

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

Rule 3552

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

Rule 3529

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

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 209

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

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 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])

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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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]

Rubi steps

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

Mathematica [C]  time = 1.48743, size = 201, normalized size = 0.63 \[ \frac{i \sqrt [3]{\tan (c+d x)} \csc (c+d x) \sec (c+d x) \left (2 \left (13 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (i \sin (2 (c+d x))+\cos (2 (c+d x))-1)+8 i \sin (2 (c+d x))+6 \cos (2 (c+d x))+6\right )-3\ 2^{2/3} \left (-1+e^{2 i (c+d x)}\right ) \sqrt [3]{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left (1-e^{2 i (c+d x)}\right )\right )\right )}{16 a d (\tan (c+d x)-i)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(5/3)*(a + I*a*Tan[c + d*x])),x]

[Out]

((I/16)*Csc[c + d*x]*Sec[c + d*x]*(-3*2^(2/3)*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hyper
geometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2] + 2*(6 + 6*Cos[2*(c + d*x)] + 13*Hypergeometric2F1[1/
3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)
]) + (8*I)*Sin[2*(c + d*x)]))*Tan[c + d*x]^(1/3))/(a*d*(-I + Tan[c + d*x]))

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 273, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{6}}}{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{1}{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{13}{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{{\frac{13\,i}{12}}\sqrt{3}}{ad}{\it Artanh} \left ({\frac{\sqrt{3}}{3} \left ( -i+2\,\sqrt [3]{\tan \left ( dx+c \right ) } \right ) } \right ) }-{\frac{1}{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{{\frac{i}{4}}\sqrt{3}}{ad}{\it Artanh} \left ({\frac{\sqrt{3}}{3} \left ( i+2\,\sqrt [3]{\tan \left ( dx+c \right ) } \right ) } \right ) }+{\frac{1}{4\,ad}\ln \left ( \sqrt [3]{\tan \left ( dx+c \right ) }-i \right ) }-{\frac{{\frac{i}{6}}}{ad} \left ( \sqrt [3]{\tan \left ( dx+c \right ) }+i \right ) ^{-1}}+{\frac{13}{12\,ad}\ln \left ( \sqrt [3]{\tan \left ( dx+c \right ) }+i \right ) }-{\frac{3}{2\,ad} \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(5/3)/(a+I*a*tan(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

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(1/tan(d*x+c)^(5/3)/(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [B]  time = 2.94622, size = 1972, normalized size = 6.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/3)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/96*((sqrt(3)*(12*I*a*d*e^(4*I*d*x + 4*I*c) - 12*I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - 12*e^(4*I*d*x
+ 4*I*c) + 12*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) + (sqrt(3)*(-12*I*a*d*e^(4*I*d*x + 4*I*c) + 12*I*a*d*e^(2*I*d*x + 2*I*c)
)*sqrt(1/(a^2*d^2)) - 12*e^(4*I*d*x + 4*I*c) + 12*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) + (sqrt(1/3)*(-156*I*a*d*e^(4*I*d*x
+ 4*I*c) + 156*I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - 52*e^(4*I*d*x + 4*I*c) + 52*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) + (sqrt(1/3)*(156*I*a*d*e^(4*I*d*x + 4*I*c) - 156*I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - 52*e^(4*
I*d*x + 4*I*c) + 52*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) + 104*(e^(4*I*d*x + 4*I*c) - 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) + 24*(e^(4*I*d*x + 4*I*c) - 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) + 4*((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I
*d*x + 2*I*c) + 1))^(1/3)*(-42*I*e^(4*I*d*x + 4*I*c) - 36*I*e^(2*I*d*x + 2*I*c) + 6*I))/(a*d*e^(4*I*d*x + 4*I*
c) - a*d*e^(2*I*d*x + 2*I*c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{i \tan ^{\frac{8}{3}}{\left (c + d x \right )} + \tan ^{\frac{5}{3}}{\left (c + d x \right )}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(5/3)/(a+I*a*tan(d*x+c)),x)

[Out]

Integral(1/(I*tan(c + d*x)**(8/3) + tan(c + d*x)**(5/3)), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.22026, size = 312, normalized size = 0.97 \begin{align*} \frac{13 i \, \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{i \, \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{\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 \, \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 \, \log \left (\tan \left (d x + c\right )^{\frac{1}{3}} + i\right )}{12 \, a d} + \frac{\log \left (\tan \left (d x + c\right )^{\frac{1}{3}} - i\right )}{4 \, a d} - \frac{3}{2 \, a d \tan \left (d x + c\right )^{\frac{2}{3}}} - \frac{\tan \left (d x + c\right )^{\frac{1}{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(1/tan(d*x+c)^(5/3)/(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

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