\(\int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx\) [240]

Optimal result

Integrand size = 26, antiderivative size = 379 \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {121 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {14 i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {121 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]

[Out]

121/72*arctan(-3^(1/2)+2*tan(d*x+c)^(1/3))/a^2/d+121/72*arctan(3^(1/2)+2*tan(d*x+c)^(1/3))/a^2/d+121/36*arctan
(tan(d*x+c)^(1/3))/a^2/d+14/9*I*ln(1+tan(d*x+c)^(2/3))/a^2/d-7/9*I*ln(1-tan(d*x+c)^(2/3)+tan(d*x+c)^(4/3))/a^2
/d-14/9*I*arctan(1/3*(1-2*tan(d*x+c)^(2/3))*3^(1/2))/a^2/d*3^(1/2)+121/144*ln(1-3^(1/2)*tan(d*x+c)^(1/3)+tan(d
*x+c)^(2/3))/a^2/d*3^(1/2)-121/144*ln(1+3^(1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))/a^2/d*3^(1/2)-14/3*I*tan(d*
x+c)^(2/3)/a^2/d-121/60*tan(d*x+c)^(5/3)/a^2/d+7/6*I*tan(d*x+c)^(8/3)/a^2/d/(1+I*tan(d*x+c))-1/4*tan(d*x+c)^(1
1/3)/d/(a+I*a*tan(d*x+c))^2

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {3639, 3676, 3609, 3619, 3557, 335, 281, 206, 31, 648, 632, 210, 642, 301, 209} \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {14 i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {121 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}+\frac {121 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}+\frac {14 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {121 \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]

[In]

Int[Tan[c + d*x]^(14/3)/(a + I*a*Tan[c + d*x])^2,x]

[Out]

(-121*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) + (121*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^
2*d) - (((14*I)/3)*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(Sqrt[3]*a^2*d) + (121*ArcTan[Tan[c + d*x]^(1/3
)])/(36*a^2*d) + (((14*I)/9)*Log[1 + Tan[c + d*x]^(2/3)])/(a^2*d) + (121*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) +
Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (121*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*S
qrt[3]*a^2*d) - (((7*I)/9)*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(a^2*d) - (((14*I)/3)*Tan[c + d*x
]^(2/3))/(a^2*d) - (121*Tan[c + d*x]^(5/3))/(60*a^2*d) + (((7*I)/6)*Tan[c + d*x]^(8/3))/(a^2*d*(1 + I*Tan[c +
d*x])) - Tan[c + d*x]^(11/3)/(4*d*(a + I*a*Tan[c + d*x])^2)

Rule 31

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

Rule 206

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 209

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

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 281

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 301

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)/(a*n*s^m))*Int[1/(r^2 + s^2*x^2), x] +
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 335

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 632

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 642

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 648

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 3557

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 3609

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 3619

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 3639

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Dist[1/(2*a^2*m), Int[(
a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)
) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*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] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m
, 2*n])

Rule 3676

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {\int \frac {\tan ^{\frac {8}{3}}(c+d x) \left (-\frac {11 a}{3}+\frac {17}{3} i a \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{4 a^2} \\ & = \frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \tan ^{\frac {5}{3}}(c+d x) \left (-\frac {224 i a^2}{9}-\frac {242}{9} a^2 \tan (c+d x)\right ) \, dx}{8 a^4} \\ & = -\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \tan ^{\frac {2}{3}}(c+d x) \left (\frac {242 a^2}{9}-\frac {224}{9} i a^2 \tan (c+d x)\right ) \, dx}{8 a^4} \\ & = -\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\frac {224 i a^2}{9}+\frac {242}{9} a^2 \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{8 a^4} \\ & = -\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(28 i) \int \frac {1}{\sqrt [3]{\tan (c+d x)}} \, dx}{9 a^2}+\frac {121 \int \tan ^{\frac {2}{3}}(c+d x) \, dx}{36 a^2} \\ & = -\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(28 i) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{9 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{36 a^2 d} \\ & = -\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(28 i) \text {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{3 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{12 a^2 d} \\ & = -\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(14 i) \text {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {121 \text {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 )}{36 a^2 d}+\frac {121 \text {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 )}{36 a^2 d} \\ & = \frac {121 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(14 i) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {(14 i) \text {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{144 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{144 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{48 \sqrt {3} a^2 d}-\frac {121 \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{48 \sqrt {3} a^2 d} \\ & = \frac {121 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {(7 i) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d}-\frac {121 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {121 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d} \\ & = -\frac {121 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(14 i) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d} \\ & = -\frac {121 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {14 i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {121 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.80 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.69 \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\tan ^{\frac {17}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\frac {56 i \arctan \left (\frac {-1+2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} d}+\frac {56 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 d}-\frac {28 i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 d}+\frac {121 i \log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{18 d \sqrt [3]{\tan (c+d x)}}-\frac {121 i \log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{18 d \sqrt [3]{\tan (c+d x)}}-\frac {121 (-1)^{5/6} \log \left (1-e^{-\frac {i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{18 d \sqrt [3]{\tan (c+d x)}}+\frac {121 \sqrt [6]{-1} \log \left (1-e^{\frac {i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{18 d \sqrt [3]{\tan (c+d x)}}-\frac {121 \sqrt [6]{-1} \log \left (1-e^{-\frac {5 i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{18 d \sqrt [3]{\tan (c+d x)}}+\frac {121 (-1)^{5/6} \log \left (1-e^{\frac {5 i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{18 d \sqrt [3]{\tan (c+d x)}}-\frac {56 i a \tan ^{\frac {2}{3}}(c+d x)}{3 d (a+i a \tan (c+d x))}+\frac {53 a \tan ^{\frac {5}{3}}(c+d x)}{5 d (a+i a \tan (c+d x))}-\frac {17 i a \tan ^{\frac {8}{3}}(c+d x)}{5 d (a+i a \tan (c+d x))}-\frac {a \tan ^{\frac {11}{3}}(c+d x)}{d (a+i a \tan (c+d x))}+\frac {i a \tan ^{\frac {14}{3}}(c+d x)}{d (a+i a \tan (c+d x))}}{4 a^2} \]

[In]

Integrate[Tan[c + d*x]^(14/3)/(a + I*a*Tan[c + d*x])^2,x]

[Out]

Tan[c + d*x]^(17/3)/(4*d*(a + I*a*Tan[c + d*x])^2) + ((((56*I)/3)*ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]])
/(Sqrt[3]*d) + (((56*I)/9)*Log[1 + Tan[c + d*x]^(2/3)])/d - (((28*I)/9)*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d
*x]^(4/3)])/d + (((121*I)/18)*Log[1 - I*(Tan[c + d*x]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6))/(d*Tan[c + d*x]^(1/3))
- (((121*I)/18)*Log[1 + I*(Tan[c + d*x]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6))/(d*Tan[c + d*x]^(1/3)) - (121*(-1)^
(5/6)*Log[1 - (Tan[c + d*x]^2)^(1/6)/E^((I/6)*Pi)]*(Tan[c + d*x]^2)^(1/6))/(18*d*Tan[c + d*x]^(1/3)) + (121*(-
1)^(1/6)*Log[1 - E^((I/6)*Pi)*(Tan[c + d*x]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6))/(18*d*Tan[c + d*x]^(1/3)) - (121
*(-1)^(1/6)*Log[1 - (Tan[c + d*x]^2)^(1/6)/E^(((5*I)/6)*Pi)]*(Tan[c + d*x]^2)^(1/6))/(18*d*Tan[c + d*x]^(1/3))
+ (121*(-1)^(5/6)*Log[1 - E^(((5*I)/6)*Pi)*(Tan[c + d*x]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6))/(18*d*Tan[c + d*x]
^(1/3)) - (((56*I)/3)*a*Tan[c + d*x]^(2/3))/(d*(a + I*a*Tan[c + d*x])) + (53*a*Tan[c + d*x]^(5/3))/(5*d*(a + I
*a*Tan[c + d*x])) - (((17*I)/5)*a*Tan[c + d*x]^(8/3))/(d*(a + I*a*Tan[c + d*x])) - (a*Tan[c + d*x]^(11/3))/(d*
(a + I*a*Tan[c + d*x])) + (I*a*Tan[c + d*x]^(14/3))/(d*(a + I*a*Tan[c + d*x])))/(4*a^2)

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(\frac {-\frac {3 \left (\tan ^{\frac {5}{3}}\left (d x +c \right )\right )}{5}-3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )+\frac {i}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {233 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {23}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}-\frac {92 \tan \left (d x +c \right )-136 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-130 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+44 i}{72 {\left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}^{2}}-\frac {233 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {233 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}}{d \,a^{2}}\)	\(246\)
default	\(\frac {-\frac {3 \left (\tan ^{\frac {5}{3}}\left (d x +c \right )\right )}{5}-3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )+\frac {i}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {233 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {23}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}-\frac {92 \tan \left (d x +c \right )-136 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-130 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+44 i}{72 {\left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}^{2}}-\frac {233 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {233 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}}{d \,a^{2}}\)	\(246\)

[In]

int(tan(d*x+c)^(14/3)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d/a^2*(-3/5*tan(d*x+c)^(5/3)-3*I*tan(d*x+c)^(2/3)+1/36*I/(tan(d*x+c)^(1/3)+I)^2+233/72*I*ln(tan(d*x+c)^(1/3)
+I)-23/36/(tan(d*x+c)^(1/3)+I)+1/16*I*ln(I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-1/8*3^(1/2)*arctanh(1/3*(I+2*t
an(d*x+c)^(1/3))*3^(1/2))-1/8*I*ln(tan(d*x+c)^(1/3)-I)-1/72*(92*tan(d*x+c)-136*I*tan(d*x+c)^(2/3)-130*tan(d*x+
c)^(1/3)+44*I)/(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)^2-233/144*I*ln(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)
-233/72*3^(1/2)*arctanh(1/3*(-I+2*tan(d*x+c)^(1/3))*3^(1/2)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (298) = 596\).

Time = 0.28 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.80 \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {45 \, {\left (\sqrt {3} {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} - i \, e^{\left (6 i \, d x + 6 i \, c\right )} - i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\frac {1}{2} \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - 45 \, {\left (\sqrt {3} {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + i \, e^{\left (6 i \, d x + 6 i \, c\right )} + i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (-\frac {1}{2} \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + 1165 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + i \, e^{\left (6 i \, d x + 6 i \, c\right )} + i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\frac {3}{2} \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) - 1165 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} - i \, e^{\left (6 i \, d x + 6 i \, c\right )} - i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (-\frac {3}{2} \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + 2330 \, {\left (-i \, e^{\left (6 i \, d x + 6 i \, c\right )} - i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + i\right ) + 90 \, {\left (i \, e^{\left (6 i \, d x + 6 i \, c\right )} + i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - i\right ) + 3 \, \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (791 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1279 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 185 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i\right )}}{720 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]

[In]

integrate(tan(d*x+c)^(14/3)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/720*(45*(sqrt(3)*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(1/(a^4*d^2)) - I*e^(6*I*d*x +
6*I*c) - I*e^(4*I*d*x + 4*I*c))*log(1/2*sqrt(3)*a^2*d*sqrt(1/(a^4*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) - 45*(sqrt(3)*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqr
t(1/(a^4*d^2)) + I*e^(6*I*d*x + 6*I*c) + I*e^(4*I*d*x + 4*I*c))*log(-1/2*sqrt(3)*a^2*d*sqrt(1/(a^4*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) + 1165*(3*sqrt(1/3)*(a^2*d*e^(6*I*d*x + 6
*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(1/(a^4*d^2)) + I*e^(6*I*d*x + 6*I*c) + I*e^(4*I*d*x + 4*I*c))*log(3/2*
sqrt(1/3)*a^2*d*sqrt(1/(a^4*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) -
1165*(3*sqrt(1/3)*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(1/(a^4*d^2)) - I*e^(6*I*d*x + 6
*I*c) - I*e^(4*I*d*x + 4*I*c))*log(-3/2*sqrt(1/3)*a^2*d*sqrt(1/(a^4*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) + 2330*(-I*e^(6*I*d*x + 6*I*c) - I*e^(4*I*d*x + 4*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) + 90*(I*e^(6*I*d*x + 6*I*c) + I*e^(4*I*d*x + 4*I*c))*l
og(((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - I) + 3*((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2
*I*d*x + 2*I*c) + 1))^(2/3)*(791*I*e^(6*I*d*x + 6*I*c) + 1279*I*e^(4*I*d*x + 4*I*c) + 185*I*e^(2*I*d*x + 2*I*c
) - 15*I))/(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(tan(d*x+c)**(14/3)/(a+I*a*tan(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(tan(d*x+c)^(14/3)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.75 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.71 \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {233 \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{144 \, a^{2} d} + \frac {\sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{16 \, a^{2} d} + \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{16 \, a^{2} d} - \frac {233 i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{144 \, a^{2} d} + \frac {233 i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{72 \, a^{2} d} - \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{8 \, a^{2} d} - \frac {23 \, \tan \left (d x + c\right )^{\frac {5}{3}} - 20 i \, \tan \left (d x + c\right )^{\frac {2}{3}}}{12 \, a^{2} d {\left (\tan \left (d x + c\right ) - i\right )}^{2}} - \frac {3 \, {\left (a^{8} d^{4} \tan \left (d x + c\right )^{\frac {5}{3}} + 5 i \, a^{8} d^{4} \tan \left (d x + c\right )^{\frac {2}{3}}\right )}}{5 \, a^{10} d^{5}} \]

[In]

integrate(tan(d*x+c)^(14/3)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")

[Out]

233/144*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) + I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) - I))/(a^2*d) + 1/16
*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) - I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) + I))/(a^2*d) + 1/16*I*log(
tan(d*x + c)^(2/3) + I*tan(d*x + c)^(1/3) - 1)/(a^2*d) - 233/144*I*log(tan(d*x + c)^(2/3) - I*tan(d*x + c)^(1/
3) - 1)/(a^2*d) + 233/72*I*log(tan(d*x + c)^(1/3) + I)/(a^2*d) - 1/8*I*log(tan(d*x + c)^(1/3) - I)/(a^2*d) - 1
/12*(23*tan(d*x + c)^(5/3) - 20*I*tan(d*x + c)^(2/3))/(a^2*d*(tan(d*x + c) - I)^2) - 3/5*(a^8*d^4*tan(d*x + c)
^(5/3) + 5*I*a^8*d^4*tan(d*x + c)^(2/3))/(a^10*d^5)

Mupad [B] (verification not implemented)

Time = 6.37 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.78 \[ \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

int(tan(c + d*x)^(14/3)/(a + a*tan(c + d*x)*1i)^2,x)

[Out]

log(((a^6*d^3*1619208448i)/3 - a^8*d^4*tan(c + d*x)^(1/3)*(1i/(512*a^6*d^3))^(1/3)*167024640i)*(1i/(512*a^6*d^
3))^(2/3) + (24321472*a^2*d*tan(c + d*x)^(1/3))/3)*(1i/(512*a^6*d^3))^(1/3) + log(((a^6*d^3*1619208448i)/3 - a
^8*d^4*tan(c + d*x)^(1/3)*(-12649337i/(373248*a^6*d^3))^(1/3)*167024640i)*(-12649337i/(373248*a^6*d^3))^(2/3)
+ (24321472*a^2*d*tan(c + d*x)^(1/3))/3)*(-12649337i/(373248*a^6*d^3))^(1/3) - ((5*tan(c + d*x)^(2/3))/(3*a^2*
d) + (tan(c + d*x)^(5/3)*23i)/(12*a^2*d))/(2*tan(c + d*x) + tan(c + d*x)^2*1i - 1i) - (tan(c + d*x)^(2/3)*3i)/
(a^2*d) - (3*tan(c + d*x)^(5/3))/(5*a^2*d) + (log((24321472*a^2*d*tan(c + d*x)^(1/3))/3 + ((3^(1/2)*1i - 1)^2*
((a^6*d^3*1619208448i)/3 - a^8*d^4*tan(c + d*x)^(1/3)*(3^(1/2)*1i - 1)*(1i/(512*a^6*d^3))^(1/3)*83512320i)*(1i
/(512*a^6*d^3))^(2/3))/4)*(3^(1/2)*1i - 1)*(1i/(512*a^6*d^3))^(1/3))/2 - (log((24321472*a^2*d*tan(c + d*x)^(1/
3))/3 + ((3^(1/2)*1i + 1)^2*((a^6*d^3*1619208448i)/3 + a^8*d^4*tan(c + d*x)^(1/3)*(3^(1/2)*1i + 1)*(1i/(512*a^
6*d^3))^(1/3)*83512320i)*(1i/(512*a^6*d^3))^(2/3))/4)*(3^(1/2)*1i + 1)*(1i/(512*a^6*d^3))^(1/3))/2 + (log((243
21472*a^2*d*tan(c + d*x)^(1/3))/3 + ((3^(1/2)*1i - 1)^2*((a^6*d^3*1619208448i)/3 - a^8*d^4*tan(c + d*x)^(1/3)*
(3^(1/2)*1i - 1)*(-12649337i/(373248*a^6*d^3))^(1/3)*83512320i)*(-12649337i/(373248*a^6*d^3))^(2/3))/4)*(3^(1/
2)*1i - 1)*(-12649337i/(373248*a^6*d^3))^(1/3))/2 - (log((24321472*a^2*d*tan(c + d*x)^(1/3))/3 + ((3^(1/2)*1i
+ 1)^2*((a^6*d^3*1619208448i)/3 + a^8*d^4*tan(c + d*x)^(1/3)*(3^(1/2)*1i + 1)*(-12649337i/(373248*a^6*d^3))^(1
/3)*83512320i)*(-12649337i/(373248*a^6*d^3))^(2/3))/4)*(3^(1/2)*1i + 1)*(-12649337i/(373248*a^6*d^3))^(1/3))/2