3.4.6 \(\int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\) [306]

Optimal. Leaf size=354 \[ \frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {4 i \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}} \]

[Out]

1/16*x*2^(2/3)/a^(4/3)-1/16*I*ln(cos(d*x+c))*2^(2/3)/a^(4/3)/d+2/3*I*ln(tan(d*x+c))/a^(4/3)/d-2*I*ln(a^(1/3)-(
a+I*a*tan(d*x+c))^(1/3))/a^(4/3)/d-3/16*I*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))*2^(2/3)/a^(4/3)/d-4/3*I
*arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))/a^(4/3)/d*3^(1/2)-1/8*I*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)*2^(2/3)/a^(4/3)/d-11/8*I/d/(a+I*a*tan(d*x+c))^(4/3)
-cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(4/3)-19/4*I/a/d/(a+I*a*tan(d*x+c))^(1/3)

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Rubi [A]
time = 0.51, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3642, 3677, 3681, 3562, 57, 631, 210, 31, 3680} \begin {gather*} -\frac {4 i \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(8*2^(1/3)*a^(4/3)) - ((4*I)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*
a^(4/3)*d) - ((I/4)*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/
3)*a^(4/3)*d) - ((I/8)*Log[Cos[c + d*x]])/(2^(1/3)*a^(4/3)*d) + (((2*I)/3)*Log[Tan[c + d*x]])/(a^(4/3)*d) - ((
2*I)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(a^(4/3)*d) - (((3*I)/8)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[
c + d*x])^(1/3)])/(2^(1/3)*a^(4/3)*d) - ((11*I)/8)/(d*(a + I*a*Tan[c + d*x])^(4/3)) - Cot[c + d*x]/(d*(a + I*a
*Tan[c + d*x])^(4/3)) - ((19*I)/4)/(a*d*(a + I*a*Tan[c + d*x])^(1/3))

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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 3677

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*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*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] && LtQ[m, 0] &&  !GtQ[n,
0]

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*} \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx &=-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}+\frac {\int \frac {\cot (c+d x) \left (-\frac {4 i a}{3}-\frac {7}{3} a \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{4/3}} \, dx}{a}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \int \frac {\cot (c+d x) \left (-\frac {32 i a^2}{9}-\frac {44}{9} a^2 \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{8 a^3}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {9 \int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (-\frac {64 i a^3}{27}-\frac {76}{27} a^3 \tan (c+d x)\right ) \, dx}{16 a^5}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {(4 i) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{2/3} \, dx}{3 a^3}-\frac {\int (a+i a \tan (c+d x))^{2/3} \, dx}{4 a^2}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {i \text {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{4 a d}-\frac {(4 i) \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{3 a d}\\ &=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}+\frac {(3 i) \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 )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {(3 i) \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 )}{8 a d}-\frac {(2 i) \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 )}{a d}\\ &=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {(4 i) \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 )}{a^{4/3} d}+\frac {(3 i) \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 )}{4 \sqrt [3]{2} a^{4/3} d}\\ &=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {4 i \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 2.07, size = 233, normalized size = 0.66 \begin {gather*} \frac {3+63 e^{2 i (c+d x)}-35 e^{4 i (c+d x)}-95 e^{6 i (c+d x)}+6 e^{4 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )+64 e^{4 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )}{8 a d \left (-1+e^{2 i (c+d x)}\right ) \left (1+e^{2 i (c+d x)}\right )^2 (-i+\tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3 + 63*E^((2*I)*(c + d*x)) - 35*E^((4*I)*(c + d*x)) - 95*E^((6*I)*(c + d*x)) + 6*E^((4*I)*(c + d*x))*(-1 + E^
((2*I)*(c + d*x)))*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))] + 64*E^((4*I)
*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c
+ d*x)))])/(8*a*d*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*(a + I*a*Tan[c +
d*x])^(1/3))

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Maple [F]
time = 0.46, size = 0, normalized size = 0.00 \[\int \frac {\cot ^{2}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 310, normalized size = 0.88 \begin {gather*} -\frac {i \, a {\left (\frac {6 \, {\left (38 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 27 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a - 3 \, a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{3}} a^{2} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a^{3}} + \frac {6 \, \sqrt {3} 2^{\frac {2}{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 )}{a^{\frac {7}{3}}} - \frac {3 \cdot 2^{\frac {2}{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 )}{a^{\frac {7}{3}}} + \frac {6 \cdot 2^{\frac {2}{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 {7}{3}}} + \frac {64 \, \sqrt {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 )}{a^{\frac {7}{3}}} - \frac {32 \, \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 )}{a^{\frac {7}{3}}} + \frac {64 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {7}{3}}}\right )}}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*I*a*(6*(38*(I*a*tan(d*x + c) + a)^2 - 27*(I*a*tan(d*x + c) + a)*a - 3*a^2)/((I*a*tan(d*x + c) + a)^(7/3)
*a^2 - (I*a*tan(d*x + c) + a)^(4/3)*a^3) + 6*sqrt(3)*2^(2/3)*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))/a^(7/3) - 3*2^(2/3)*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))/a^(7/3) + 6*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) +
a)^(1/3))/a^(7/3) + 64*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3)
- 32*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))/a^(7/3) + 64*log((I*a*
tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(7/3))/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (252) = 504\).
time = 1.59, size = 792, normalized size = 2.24 \begin {gather*} \frac {2^{\frac {2}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (-95 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 35 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 63 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {4}{3} i \, d x + \frac {4}{3} i \, c\right )} + 32 \, {\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 )} \left (\frac {64 i}{27 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {9}{16} \, a^{3} d^{2} \left (\frac {64 i}{27 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \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 )}\right ) + 32 \, {\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 )} \left (\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (32 \, a^{3} d^{2} \left (\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \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 )}\right ) - 16 \, {\left ({\left (-i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (i \, \sqrt {3} a^{2} d - a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (\frac {64 i}{27 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {9}{32} \, {\left (i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (\frac {64 i}{27 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \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 )}\right ) - 16 \, {\left ({\left (i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-i \, \sqrt {3} a^{2} d - a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (\frac {64 i}{27 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {9}{32} \, {\left (-i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (\frac {64 i}{27 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \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 )}\right ) - 16 \, {\left ({\left (-i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (i \, \sqrt {3} a^{2} d - a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-16 \, {\left (i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \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 )}\right ) - 16 \, {\left ({\left (i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-i \, \sqrt {3} a^{2} d - a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-16 \, {\left (-i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \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 )}\right )}{32 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(-95*I*e^(6*I*d*x + 6*I*c) - 35*I*e^(4*I*d*x + 4*I*c) + 63*I
*e^(2*I*d*x + 2*I*c) + 3*I)*e^(4/3*I*d*x + 4/3*I*c) + 32*(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c
))*(64/27*I/(a^4*d^3))^(1/3)*log(9/16*a^3*d^2*(64/27*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1)
)^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 32*(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))*(1/128*I/(a^4*d^
3))^(1/3)*log(32*a^3*d^2*(1/128*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x
+ 2/3*I*c)) - 16*((-I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) + (I*sqrt(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*
c))*(64/27*I/(a^4*d^3))^(1/3)*log(-9/32*(I*sqrt(3)*a^3*d^2 + a^3*d^2)*(64/27*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(
e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 16*((I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) +
(-I*sqrt(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*c))*(64/27*I/(a^4*d^3))^(1/3)*log(-9/32*(-I*sqrt(3)*a^3*d^2 + a^3
*d^2)*(64/27*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 16*((
-I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) + (I*sqrt(3)*a^2*d - a^2*d)*e^(4*I*d*x + 4*I*c))*(1/128*I/(a^4*d
^3))^(1/3)*log(-16*(I*sqrt(3)*a^3*d^2 + a^3*d^2)*(1/128*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) +
1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 16*((I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) + (-I*sqrt(3)*a^2*d -
a^2*d)*e^(4*I*d*x + 4*I*c))*(1/128*I/(a^4*d^3))^(1/3)*log(-16*(-I*sqrt(3)*a^3*d^2 + a^3*d^2)*(1/128*I/(a^4*d^3
))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)))/(a^2*d*e^(6*I*d*x + 6*I*c) -
a^2*d*e^(4*I*d*x + 4*I*c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.99, size = 893, normalized size = 2.52 \begin {gather*} -\frac {\frac {\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,27{}\mathrm {i}}{8\,d}+\frac {a\,3{}\mathrm {i}}{8\,d}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2\,19{}\mathrm {i}}{4\,a\,d}}{a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}-{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/3}}+\ln \left (d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,1584{}\mathrm {i}-\left (\left (46656\,a^7\,d^6\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{2/3}+55782\,a^4\,d^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{1/3}-a^3\,d^3\,37107{}\mathrm {i}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{1/3}+\ln \left (d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,1584{}\mathrm {i}-\left (\left (46656\,a^7\,d^6\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{2/3}+55782\,a^4\,d^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{1/3}-a^3\,d^3\,37107{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{1/3}+\frac {\ln \left (d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,1584{}\mathrm {i}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,37107{}\mathrm {i}-\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (55782\,a^4\,d^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+11664\,a^7\,d^6\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{2/3}}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,1584{}\mathrm {i}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,37107{}\mathrm {i}+\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (55782\,a^4\,d^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+11664\,a^7\,d^6\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{2/3}}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {64{}\mathrm {i}}{27\,a^4\,d^3}\right )}^{1/3}}{2}+\frac {\ln \left (d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,1584{}\mathrm {i}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,37107{}\mathrm {i}-\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (55782\,a^4\,d^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+11664\,a^7\,d^6\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{2/3}}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,1584{}\mathrm {i}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,37107{}\mathrm {i}+\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (55782\,a^4\,d^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+11664\,a^7\,d^6\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{2/3}}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{128\,a^4\,d^3}\right )}^{1/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i - ((46656*a^7*d^6*(64i/(27*a^4*d^3))^(2/3) + 55782*a^4*d^4*(a + a*ta
n(c + d*x)*1i)^(1/3))*(64i/(27*a^4*d^3))^(1/3) - a^3*d^3*37107i)*(64i/(27*a^4*d^3))^(2/3))*(64i/(27*a^4*d^3))^
(1/3) - (((a + a*tan(c + d*x)*1i)*27i)/(8*d) + (a*3i)/(8*d) - ((a + a*tan(c + d*x)*1i)^2*19i)/(4*a*d))/(a*(a +
a*tan(c + d*x)*1i)^(4/3) - (a + a*tan(c + d*x)*1i)^(7/3)) + log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i - ((466
56*a^7*d^6*(1i/(128*a^4*d^3))^(2/3) + 55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3))*(1i/(128*a^4*d^3))^(1/3) -
a^3*d^3*37107i)*(1i/(128*a^4*d^3))^(2/3))*(1i/(128*a^4*d^3))^(1/3) + (log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584
i + ((3^(1/2)*1i - 1)^2*(a^3*d^3*37107i - ((3^(1/2)*1i - 1)*(55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 116
64*a^7*d^6*(3^(1/2)*1i - 1)^2*(64i/(27*a^4*d^3))^(2/3))*(64i/(27*a^4*d^3))^(1/3))/2)*(64i/(27*a^4*d^3))^(2/3))
/4)*(3^(1/2)*1i - 1)*(64i/(27*a^4*d^3))^(1/3))/2 - (log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i + ((3^(1/2)*1i +
1)^2*(a^3*d^3*37107i + ((3^(1/2)*1i + 1)*(55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 11664*a^7*d^6*(3^(1/2
)*1i + 1)^2*(64i/(27*a^4*d^3))^(2/3))*(64i/(27*a^4*d^3))^(1/3))/2)*(64i/(27*a^4*d^3))^(2/3))/4)*(3^(1/2)*1i +
1)*(64i/(27*a^4*d^3))^(1/3))/2 + (log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i + ((3^(1/2)*1i - 1)^2*(a^3*d^3*371
07i - ((3^(1/2)*1i - 1)*(55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 11664*a^7*d^6*(3^(1/2)*1i - 1)^2*(1i/(1
28*a^4*d^3))^(2/3))*(1i/(128*a^4*d^3))^(1/3))/2)*(1i/(128*a^4*d^3))^(2/3))/4)*(3^(1/2)*1i - 1)*(1i/(128*a^4*d^
3))^(1/3))/2 - (log(d*(a + a*tan(c + d*x)*1i)^(1/3)*1584i + ((3^(1/2)*1i + 1)^2*(a^3*d^3*37107i + ((3^(1/2)*1i
+ 1)*(55782*a^4*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 11664*a^7*d^6*(3^(1/2)*1i + 1)^2*(1i/(128*a^4*d^3))^(2/3)
)*(1i/(128*a^4*d^3))^(1/3))/2)*(1i/(128*a^4*d^3))^(2/3))/4)*(3^(1/2)*1i + 1)*(1i/(128*a^4*d^3))^(1/3))/2

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