\(\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\) [285]

Optimal result

Integrand size = 26, antiderivative size = 315 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {a^{4/3} x}{2^{2/3}}-\frac {4 i a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}+\frac {i \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac {2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac {3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \]

[Out]

1/2*a^(4/3)*x*2^(1/3)-1/2*I*a^(4/3)*ln(cos(d*x+c))*2^(1/3)/d-2/3*I*a^(4/3)*ln(tan(d*x+c))/d+2*I*a^(4/3)*ln(a^(
1/3)-(a+I*a*tan(d*x+c))^(1/3))/d-3/2*I*a^(4/3)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))*2^(1/3)/d-4/3*I*a^
(4/3)*arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))/d*3^(1/2)+I*2^(1/3)*a^(4/3)*arctan(1/3*
(a^(1/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)/d+I*a*(a+I*a*tan(d*x+c))^(1/3)/d-cot(d*x+c
)*(a+I*a*tan(d*x+c))^(4/3)/d

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3642, 3675, 3681, 3562, 59, 631, 210, 31, 3680} \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {4 i a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}+\frac {i \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac {2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac {3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac {a^{4/3} x}{2^{2/3}}+\frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \]

[In]

Int[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3),x]

[Out]

(a^(4/3)*x)/2^(2/3) - ((4*I)*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*d) + (I*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3)
)])/d - (I*a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (((2*I)/3)*a^(4/3)*Log[Tan[c + d*x]])/d + ((2*I)*a^(4/3)*L
og[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/d - ((3*I)*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1
/3)])/(2^(2/3)*d) + (I*a*(a + I*a*Tan[c + d*x])^(1/3))/d - (Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3))/d

Rule 31

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

Rule 59

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

Rule 210

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

Rule 631

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

Rule 3562

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

Rule 3642

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

Rule 3675

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

Rule 3680

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

Rule 3681

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac {\int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \left (\frac {4 i a}{3}+\frac {1}{3} a \tan (c+d x)\right ) \, dx}{a} \\ & = \frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac {3 \int \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \left (\frac {4 i a^2}{9}-\frac {2}{9} a^2 \tan (c+d x)\right ) \, dx}{a} \\ & = \frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac {4}{3} i \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx-(2 a) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx \\ & = \frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac {\left (4 i a^2\right ) \text {Subst}\left (\int \frac {1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{3 d}+\frac {\left (2 i a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = \frac {a^{4/3} x}{2^{2/3}}-\frac {i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}-\frac {\left (2 i a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}+\frac {\left (3 i a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {\left (2 i a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}+\frac {\left (3 i a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{\sqrt [3]{2} d} \\ & = \frac {a^{4/3} x}{2^{2/3}}-\frac {i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac {2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac {3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac {\left (4 i a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{d}-\frac {\left (3 i \sqrt [3]{2} a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{d} \\ & = \frac {a^{4/3} x}{2^{2/3}}-\frac {4 i a^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} d}+\frac {i \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{d}-\frac {i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac {2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac {3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i \left (8 \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )-6 \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-8 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+6 \sqrt [3]{2} a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+4 a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-3 \sqrt [3]{2} a^{4/3} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-6 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}\right )}{6 d} \]

[In]

Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3),x]

[Out]

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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {3 i a^{3} \left (-\frac {2 \left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right )}{a}-\frac {-\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{3 a \tan \left (d x +c \right )}-\frac {4 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )}{d}\)	\(278\)
default	\(\frac {3 i a^{3} \left (-\frac {2 \left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right )}{a}-\frac {-\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{3 a \tan \left (d x +c \right )}-\frac {4 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )}{d}\)	\(278\)

[In]

int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x,method=_RETURNVERBOSE)

[Out]

3*I/d*a^3*(-2*(1/6*2^(1/3)/a^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))-1/12*2^(1/3)/a^(2/3)*ln((a+I*a
*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+2^(2/3)*a^(2/3))-1/6*2^(1/3)/a^(2/3)*3^(1/2)*arcta
n(1/3*3^(1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))/a-1/a*(-1/3*I*(a+I*a*tan(d*x+c))^(1/3)/a/tan(d*x+
c)-4/9/a^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-a^(1/3))+2/9/a^(2/3)*ln((a+I*a*tan(d*x+c))^(2/3)+a^(1/3)*(a+I*a*tan
(d*x+c))^(1/3)+a^(2/3))+4/9/a^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))))

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. 622 vs. \(2 (232) = 464\).

Time = 0.26 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.97 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {2 \cdot 2^{\frac {1}{3}} {\left (i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a\right )} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - {\left ({\left (i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {3} d + d\right )} \left (-\frac {64 i \, a^{4}}{27 \, d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {8 \cdot 2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - 3 \, {\left (\sqrt {3} d + i \, d\right )} \left (-\frac {64 i \, a^{4}}{27 \, d^{3}}\right )^{\frac {1}{3}}}{8 \, a}\right ) - {\left ({\left (-i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {3} d + d\right )} \left (-\frac {64 i \, a^{4}}{27 \, d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {8 \cdot 2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + 3 \, {\left (\sqrt {3} d - i \, d\right )} \left (-\frac {64 i \, a^{4}}{27 \, d^{3}}\right )^{\frac {1}{3}}}{8 \, a}\right ) - 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \left (-\frac {64 i \, a^{4}}{27 \, d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {4 \cdot 2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + 3 i \, \left (-\frac {64 i \, a^{4}}{27 \, d^{3}}\right )^{\frac {1}{3}} d}{4 \, a}\right ) - {\left ({\left (i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {3} d + d\right )} \left (\frac {2 i \, a^{4}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2 \cdot 2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (\sqrt {3} d + i \, d\right )} \left (\frac {2 i \, a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) - {\left ({\left (-i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {3} d + d\right )} \left (\frac {2 i \, a^{4}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2 \cdot 2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - {\left (\sqrt {3} d - i \, d\right )} \left (\frac {2 i \, a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) - 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \left (\frac {2 i \, a^{4}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - i \, \left (\frac {2 i \, a^{4}}{d^{3}}\right )^{\frac {1}{3}} d}{a}\right )}{2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

-1/2*(2*2^(1/3)*(I*a*e^(2*I*d*x + 2*I*c) + I*a)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) -
((I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/8*(8*2^(1/3)*a*(a/(e^
(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - 3*(sqrt(3)*d + I*d)*(-64/27*I*a^4/d^3)^(1/3))/a) - ((-
I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/8*(8*2^(1/3)*a*(a/(e^(2
*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + 3*(sqrt(3)*d - I*d)*(-64/27*I*a^4/d^3)^(1/3))/a) - 2*(d*
e^(2*I*d*x + 2*I*c) - d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/4*(4*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(
2/3*I*d*x + 2/3*I*c) + 3*I*(-64/27*I*a^4/d^3)^(1/3)*d)/a) - ((I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)
*d + d)*(2*I*a^4/d^3)^(1/3)*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) +
(sqrt(3)*d + I*d)*(2*I*a^4/d^3)^(1/3))/a) - ((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*(2*I*a
^4/d^3)^(1/3)*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (sqrt(3)*d -
I*d)*(2*I*a^4/d^3)^(1/3))/a) - 2*(d*e^(2*I*d*x + 2*I*c) - d)*(2*I*a^4/d^3)^(1/3)*log((2^(1/3)*a*(a/(e^(2*I*d*x
+ 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - I*(2*I*a^4/d^3)^(1/3)*d)/a))/(d*e^(2*I*d*x + 2*I*c) - d)

Sympy [F]

\[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}} \cot ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**2*(a+I*a*tan(d*x+c))**(4/3),x)

[Out]

Integral((I*a*(tan(c + d*x) - I))**(4/3)*cot(c + d*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.82 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {i \, {\left (6 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 8 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 3 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 6 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 4 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 8 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {6 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{\tan \left (d x + c\right )}\right )} a}{6 \, d} \]

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

1/6*I*(6*sqrt(3)*2^(1/3)*a^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I*a*tan(d*x + c) + a)^(1/3))
/a^(1/3)) - 8*sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3)) + 3*2^(1/
3)*a^(1/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(2/3))
- 6*2^(1/3)*a^(1/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3)) - 4*a^(1/3)*log((I*a*tan(d*x + c) + a
)^(2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3)) + 8*a^(1/3)*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/
3)) + 6*I*(I*a*tan(d*x + c) + a)^(1/3)/tan(d*x + c))*a/d

Giac [F]

\[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^(4/3)*cot(d*x + c)^2, x)

Mupad [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 855, normalized size of antiderivative = 2.71 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^2*(a + a*tan(c + d*x)*1i)^(4/3),x)

[Out]

log(((1458*a^7*d^6*((a^4*2i)/d^3)^(1/3) - a^8*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*810i)*((a^4*2i)/d^3)^(2/3) - a
^11*d^3*792i)*((a^4*2i)/d^3)^(1/3) - 3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3))*((a^4*2i)/d^3)^(1/3) + log((
(1458*a^7*d^6*(-(a^4*64i)/(27*d^3))^(1/3) - a^8*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*810i)*(-(a^4*64i)/(27*d^3))^
(2/3) - a^11*d^3*792i)*(-(a^4*64i)/(27*d^3))^(1/3) - 3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3))*(-(a^4*64i)/
(27*d^3))^(1/3) + (log(3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3) + ((3^(1/2)*1i - 1)*(a^11*d^3*792i + ((3^(1
/2)*1i - 1)^2*(a^8*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*810i - 729*a^7*d^6*(3^(1/2)*1i - 1)*((a^4*2i)/d^3)^(1/3))
*((a^4*2i)/d^3)^(2/3))/4)*((a^4*2i)/d^3)^(1/3))/2)*(3^(1/2)*1i - 1)*((a^4*2i)/d^3)^(1/3))/2 - (log(3744*a^12*d
^2*(a + a*tan(c + d*x)*1i)^(1/3) - ((3^(1/2)*1i + 1)*(a^11*d^3*792i + ((3^(1/2)*1i + 1)^2*(a^8*d^5*(a + a*tan(
c + d*x)*1i)^(1/3)*810i + 729*a^7*d^6*(3^(1/2)*1i + 1)*((a^4*2i)/d^3)^(1/3))*((a^4*2i)/d^3)^(2/3))/4)*((a^4*2i
)/d^3)^(1/3))/2)*(3^(1/2)*1i + 1)*((a^4*2i)/d^3)^(1/3))/2 + (log(3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3) +
((3^(1/2)*1i - 1)*(a^11*d^3*792i + ((3^(1/2)*1i - 1)^2*(a^8*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*810i - 729*a^7*
d^6*(3^(1/2)*1i - 1)*(-(a^4*64i)/(27*d^3))^(1/3))*(-(a^4*64i)/(27*d^3))^(2/3))/4)*(-(a^4*64i)/(27*d^3))^(1/3))
/2)*(3^(1/2)*1i - 1)*(-(a^4*64i)/(27*d^3))^(1/3))/2 - (log(3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3) - ((3^(
1/2)*1i + 1)*(a^11*d^3*792i + ((3^(1/2)*1i + 1)^2*(a^8*d^5*(a + a*tan(c + d*x)*1i)^(1/3)*810i + 729*a^7*d^6*(3
^(1/2)*1i + 1)*(-(a^4*64i)/(27*d^3))^(1/3))*(-(a^4*64i)/(27*d^3))^(2/3))/4)*(-(a^4*64i)/(27*d^3))^(1/3))/2)*(3
^(1/2)*1i + 1)*(-(a^4*64i)/(27*d^3))^(1/3))/2 - (a*(a + a*tan(c + d*x)*1i)^(1/3))/(d*tan(c + d*x))