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

Optimal. Leaf size=254 \[ -\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}+\frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{a^{4/3} \log (\tan (c+d x))}{2 d}+\frac{3 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{i a^{4/3} x}{2^{2/3}} \]

[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)

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Rubi [A]  time = 0.415998, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3562, 3478, 3481, 57, 617, 204, 31, 3594, 3599} \[ -\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}+\frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{a^{4/3} \log (\tan (c+d x))}{2 d}+\frac{3 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{i a^{4/3} x}{2^{2/3}} \]

Antiderivative was successfully verified.

[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 3562

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 3478

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 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 3594

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 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 (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx &=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 \operatorname{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 ) \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{a^{4/3} \log (\tan (c+d x))}{2 d}-\frac{\left (3 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 )}{2 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 (3 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 )}{2 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{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 ) \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 )}{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{\sqrt{3} 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 )}{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{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]  time = 3.97075, size = 411, normalized size = 1.62 \[ \frac{a e^{-\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}} \sqrt [3]{a+i a \tan (c+d x)} \left (-4 \log \left (1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )+2\ 2^{2/3} \log \left (1-\frac{\sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )-2^{2/3} \log \left (\frac{\sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}+\frac{2^{2/3} e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}+1\right )+2 \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )-2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( dx+c \right ) \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)*(a+I*a*tan(d*x+c))^(4/3),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.7537, size = 1251, normalized size = 4.93 \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)*(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)

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

[Out]

Timed out

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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 )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)