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

Optimal. Leaf size=321 \[ \frac{11 a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} d}-\frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}+\frac{11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac{11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 d}+\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{i a^{4/3} x}{2^{2/3}}-\frac{\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac{2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.541102, antiderivative size = 321, normalized size of antiderivative = 1., 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 = {3561, 3593, 3600, 3481, 57, 617, 204, 31, 3599} \[ \frac{11 a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} d}-\frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}+\frac{11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac{11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 d}+\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{i a^{4/3} x}{2^{2/3}}-\frac{\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac{2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d} \]

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 3561

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 3593

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] && EqQ
[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3600

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]

Rule 3481

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 57

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 617

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 204

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

Rule 31

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

Rule 3599

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*} \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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [F]  time = 180.005, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

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

________________________________________________________________________________________

Maple [F]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{3} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.84148, size = 2130, normalized size = 6.64 \begin{align*} \text{result too large to display} \end{align*}

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]

1/54*(18*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)*(3*(I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 6*(-I*sqrt(3)*d - d)*e^(2*I*d*
x + 2*I*c) + 3*I*sqrt(3)*d + 3*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) - 27*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) +
54*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*(3*(I*sqrt(3)*d + d)*e^(4*
I*d*x + 4*I*c) + 6*(-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + 3*I*sqrt(3)*d + 3*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)
- 33*((-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) + 66*(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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{4}{3}} \cot \left (d x + c\right )^{3}\,{d x} \end{align*}

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)