3.3.86 \(\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\) [286]

Optimal. Leaf size=321 \[ \frac {i a^{4/3} x}{2^{2/3}}+\frac {11 a^{4/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} d}-\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac {11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac {11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 d}+\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d} \]

[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+11/18*a^(4/3)*ln(tan(d*x+c))/d-11/6*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+11/9*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))*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-2/3*I*a*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/3)/
d-1/2*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3)/d

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Rubi [A]
time = 0.39, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3642, 3674, 3681, 3562, 59, 631, 210, 31, 3680} \begin {gather*} \frac {11 a^{4/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} d}-\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \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 )}{d}+\frac {11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac {11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 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}}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*a^(4/3)*x)/2^(2/3) + (11*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(3*S
qrt[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) + (11*a^(4/3)*Log[Tan[c + d*x]])/(18*d) - (11*a^(4/3)*Log[a^(1/
3) - (a + I*a*Tan[c + d*x])^(1/3)])/(6*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) - (((2*I)/3)*a*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d - (Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])
^(4/3))/(2*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 3674

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^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x]
)^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c
+ d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m
- 1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && E
qQ[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*} \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx &=-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}+\frac {\int \cot ^2(c+d x) \left (\frac {4 i a}{3}-\frac {2}{3} a \tan (c+d x)\right ) (a+i a \tan (c+d x))^{4/3} \, dx}{2 a}\\ &=-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}+\frac {\int \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \left (-\frac {22 a^2}{9}-\frac {14}{9} i a^2 \tan (c+d x)\right ) \, dx}{2 a}\\ &=-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac {11}{9} \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx-(2 i a) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac {\left (11 a^2\right ) \text {Subst}\left (\int \frac {1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{9 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 {11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}+\frac {\left (11 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 )}{6 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 (11 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 )}{6 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 {11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac {11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 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 {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac {\left (11 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 )}{3 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 {11 a^{4/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} d}-\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{d}+\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac {11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac {11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 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 {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\\ \end {align*}

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int \left (\cot ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.55, size = 304, normalized size = 0.95 \begin {gather*} -\frac {{\left (\frac {18 \, \sqrt {3} 2^{\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 )}{a^{\frac {2}{3}}} - \frac {22 \, \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 {2}{3}}} + \frac {9 \cdot 2^{\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 )}{a^{\frac {2}{3}}} - \frac {18 \cdot 2^{\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 )}{a^{\frac {2}{3}}} - \frac {11 \, \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 {2}{3}}} + \frac {22 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} - \frac {3 \, {\left (7 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + a^{2}}\right )} a^{2}}{18 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(18*sqrt(3)*2^(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
))/a^(2/3) - 22*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(2/3) + 9*2^(
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))/a^(2/3
) - 18*2^(1/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3))/a^(2/3) - 11*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^(2/3) + 22*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))
/a^(2/3) - 3*(7*(I*a*tan(d*x + c) + a)^(4/3) - 4*(I*a*tan(d*x + c) + a)^(1/3)*a)/((I*a*tan(d*x + c) + a)^2 - 2
*(I*a*tan(d*x + c) + a)*a + a^2))*a^2/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. 744 vs. \(2 (240) = 480\).
time = 1.20, size = 744, normalized size = 2.32 \begin {gather*} \frac {6 \cdot 2^{\frac {1}{3}} {\left (5 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, 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 )} - 9 \cdot 2^{\frac {1}{3}} {\left ({\left (i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {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 )} + 2^{\frac {1}{3}} {\left (i \, \sqrt {3} d + d\right )} \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) - 9 \cdot 2^{\frac {1}{3}} {\left ({\left (-i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {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 )} + 2^{\frac {1}{3}} {\left (-i \, \sqrt {3} d + d\right )} \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + 18 \cdot 2^{\frac {1}{3}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\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 )} - 2^{\frac {1}{3}} \left (\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} d}{a}\right ) - 11 \, {\left ({\left (i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {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 (i \, \sqrt {3} d + d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) - 11 \, {\left ({\left (-i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\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 {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 (-i \, \sqrt {3} d + d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + 22 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\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 )}{18 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(6*2^(1/3)*(5*a*e^(4*I*d*x + 4*I*c) + 3*a*e^(2*I*d*x + 2*I*c) - 2*a)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*
e^(2/3*I*d*x + 2/3*I*c) - 9*2^(1/3)*((I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 2*(-I*sqrt(3)*d - d)*e^(2*I*d*x +
2*I*c) + I*sqrt(3)*d + d)*(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) + 2^(1/3)*(I*sqrt(3)*d + d)*(a^4/d^3)^(1/3))/a) - 9*2^(1/3)*((-I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I
*c) + 2*(I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(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) + 2^(1/3)*(-I*sqrt(3)*d + d)*(a^4/d^3)^(1/3))/a) + 18*2^(1
/3)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(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) - 11*((I*sqrt(3)*d + d)*e^(4*I*d*x + 4*
I*c) + 2*(-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*(-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) - (I*sqrt(3)*d + d)*(-a^4/d^3)^(1/3))/a) - 11*((-I*sqrt
(3)*d + d)*e^(4*I*d*x + 4*I*c) + 2*(I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(-a^4/d^3)^(1/3)*l
og(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) + 22*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(-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))/(d*e^(4*I*d*x + 4*I*c) - 2*d*e^
(2*I*d*x + 2*I*c) + d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)^3*(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)^3, x)

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Mupad [B]
time = 4.41, size = 460, normalized size = 1.43 \begin {gather*} -\frac {\frac {2\,a^3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{3}-\frac {7\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}{6}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2+a^2\,d-2\,a\,d\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}+\frac {11\,\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}}{9}+\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 {11\,\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}}{18}+\frac {11\,\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}}{18}+\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}-\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*log(d*(-a^4/d^3)^(1/3) + a*(a*(tan(c + d*x)*1i + 1))^(1/3))*(-a^4/d^3)^(1/3))/9 - ((2*a^3*(a + a*tan(c + d
*x)*1i)^(1/3))/3 - (7*a^2*(a + a*tan(c + d*x)*1i)^(4/3))/6)/(d*(a + a*tan(c + d*x)*1i)^2 + a^2*d - 2*a*d*(a +
a*tan(c + d*x)*1i)) + 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) -
(11*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))/18 + (11*log(2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) - d*(-a^4/d^3)^(1/3) + 3^(1/2)*d*(-a^
4/d^3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(-a^4/d^3)^(1/3))/18 + 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) - 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)

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