\(\int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\) [284]

Optimal result

Integrand size = 24, antiderivative size = 254 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i a^{4/3} x}{2^{2/3}}-\frac {\sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {a^{4/3} \log (\tan (c+d x))}{2 d}+\frac {3 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3643, 3559, 3562, 59, 631, 210, 31, 3675, 3680} \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {\sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {a^{4/3} \log (\tan (c+d x))}{2 d}+\frac {3 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {i a^{4/3} x}{2^{2/3}} \]

[In]

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

[Out]

((-I)*a^(4/3)*x)/2^(2/3) - (Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3)
)])/d + (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
- (a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (a^(4/3)*Log[Tan[c + d*x]])/(2*d) + (3*a^(4/3)*Log[a^(1/3) - (a +
I*a*Tan[c + d*x])^(1/3)])/(2*d) - (3*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*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 3559

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

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 3643

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[a/(
a*c - b*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[d/(a*c - b*d), Int[(a + b*Tan[e + f*x])^m*((b + a*Tan[e
+ f*x])/(c + d*Tan[e + f*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]

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]

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.12 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {a^{4/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+2 \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-\sqrt [3]{2} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}+\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{d}-\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 d}-\frac {a^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}\)	\(245\)
default	\(-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}+\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{d}-\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 d}-\frac {a^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}\)	\(245\)

[In]

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

[Out]

-1/d*a^(4/3)*2^(1/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))+1/2/d*a^(4/3)*2^(1/3)*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/d*a^(4/3)*2^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)
*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))+1/d*a^(4/3)*ln((a+I*a*tan(d*x+c))^(1/3)-a^(1/3))-1/2/d*a^(4/3)*
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))-1/d*a^(4/3)*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. 446 vs. \(2 (188) = 376\).

Time = 0.26 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.76 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {1}{2} \cdot 2^{\frac {1}{3}} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )} \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 )} + 2^{\frac {1}{3}} {\left (i \, \sqrt {3} d - d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{3}} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )} \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 )} + 2^{\frac {1}{3}} {\left (-i \, \sqrt {3} d - d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + \frac {1}{2} \, \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )} \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 (i \, \sqrt {3} d + d\right )} \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + \frac {1}{2} \, \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )} \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 (-i \, \sqrt {3} d + d\right )} \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + 2^{\frac {1}{3}} \left (-\frac {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 )} + 2^{\frac {1}{3}} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} d}{a}\right ) + \left (\frac {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 )} - \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} d}{a}\right ) \]

[In]

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

[Out]

1/2*2^(1/3)*(-a^4/d^3)^(1/3)*(I*sqrt(3) - 1)*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) + 2^(1/3)*(I*sqrt(3)*d - d)*(-a^4/d^3)^(1/3))/a) + 1/2*2^(1/3)*(-a^4/d^3)^(1/3)*(-I*sqrt(3) -
1)*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) + 2^(1/3)*(-I*sqrt(3)*d -
d)*(-a^4/d^3)^(1/3))/a) + 1/2*(a^4/d^3)^(1/3)*(-I*sqrt(3) - 1)*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) + (I*sqrt(3)*d + d)*(a^4/d^3)^(1/3))/a) + 1/2*(a^4/d^3)^(1/3)*(I*sqrt(3) - 1
)*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) + (-I*sqrt(3)*d + d)*(a^4/d
^3)^(1/3))/a) + 2^(1/3)*(-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) + 2^(1/3)*(-a^4/d^3)^(1/3)*d)/a) + (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) - (a^4/d^3)^(1/3)*d)/a)

Sympy [F]

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

[In]

integrate(cot(d*x+c)*(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), x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.92 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 2 \, \sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2^{\frac {1}{3}} a^{\frac {4}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 2 \cdot 2^{\frac {1}{3}} a^{\frac {4}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - a^{\frac {4}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {4}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{2 \, d} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate(cot(d*x+c)*(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), x)

Mupad [B] (verification not implemented)

Time = 5.34 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.45 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\ln \left (-d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}\right )\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+\ln \left (a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+2^{1/3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}\right )\,{\left (-\frac {2\,a^4}{d^3}\right )}^{1/3}+\frac {\ln \left (d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-\sqrt {3}\,d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a^4}{d^3}\right )}^{1/3}}{2}-\frac {\ln \left (d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\sqrt {3}\,d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a^4}{d^3}\right )}^{1/3}}{2}-\ln \left (2^{1/3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}-2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,a^4}{d^3}\right )}^{1/3}+\ln \left (2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,a^4}{d^3}\right )}^{1/3} \]

[In]

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

[Out]

log(a*(a*(tan(c + d*x)*1i + 1))^(1/3) - d*(a^4/d^3)^(1/3))*(a^4/d^3)^(1/3) + log(a*(a*(tan(c + d*x)*1i + 1))^(
1/3) + 2^(1/3)*d*(-a^4/d^3)^(1/3))*(-(2*a^4)/d^3)^(1/3) + (log(d*(a^4/d^3)^(1/3) + 2*a*(a*(tan(c + d*x)*1i + 1
))^(1/3) - 3^(1/2)*d*(a^4/d^3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(a^4/d^3)^(1/3))/2 - (log(d*(a^4/d^3)^(1/3) + 2*a*(a
*(tan(c + d*x)*1i + 1))^(1/3) + 3^(1/2)*d*(a^4/d^3)^(1/3)*1i)*(3^(1/2)*1i + 1)*(a^4/d^3)^(1/3))/2 - log(2^(1/3
)*d*(-a^4/d^3)^(1/3) - 2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) + 2^(1/3)*3^(1/2)*d*(-a^4/d^3)^(1/3)*1i)*((3^(1/2)*
1i)/2 + 1/2)*(-(2*a^4)/d^3)^(1/3) + log(2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) - 2^(1/3)*d*(-a^4/d^3)^(1/3) + 2^(
1/3)*3^(1/2)*d*(-a^4/d^3)^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(-(2*a^4)/d^3)^(1/3)