3.446 \(\int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx\)

Optimal. Leaf size=37 \[ \frac{3 i a (d \sec (e+f x))^{2/3}}{f \sqrt [3]{a+i a \tan (e+f x)}} \]

[Out]

((3*I)*a*(d*Sec[e + f*x])^(2/3))/(f*(a + I*a*Tan[e + f*x])^(1/3))

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Rubi [A]  time = 0.0772428, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {3493} \[ \frac{3 i a (d \sec (e+f x))^{2/3}}{f \sqrt [3]{a+i a \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(2/3),x]

[Out]

((3*I)*a*(d*Sec[e + f*x])^(2/3))/(f*(a + I*a*Tan[e + f*x])^(1/3))

Rule 3493

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*b*
(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2
, 0] && EqQ[Simplify[m/2 + n - 1], 0]

Rubi steps

\begin{align*} \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx &=\frac{3 i a (d \sec (e+f x))^{2/3}}{f \sqrt [3]{a+i a \tan (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.324315, size = 47, normalized size = 1.27 \[ \frac{3 d^2 (\tan (e+f x)+i) (a+i a \tan (e+f x))^{2/3}}{f (d \sec (e+f x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(2/3),x]

[Out]

(3*d^2*(I + Tan[e + f*x])*(a + I*a*Tan[e + f*x])^(2/3))/(f*(d*Sec[e + f*x])^(4/3))

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{{\frac{2}{3}}} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(f*x+e))^(2/3)*(a+I*a*tan(f*x+e))^(2/3),x)

[Out]

int((d*sec(f*x+e))^(2/3)*(a+I*a*tan(f*x+e))^(2/3),x)

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Maxima [B]  time = 1.99047, size = 144, normalized size = 3.89 \begin{align*} -\frac{3 \,{\left (-i \cdot 2^{\frac{1}{3}} \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 2^{\frac{1}{3}} \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} a^{\frac{2}{3}} d^{\frac{2}{3}}}{{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac{1}{6}} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^(2/3)*(a+I*a*tan(f*x+e))^(2/3),x, algorithm="maxima")

[Out]

-3*(-I*2^(1/3)*cos(1/3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - 2^(1/3)*sin(1/3*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e) + 1)))*a^(2/3)*d^(2/3)/((cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) +
1)^(1/6)*f)

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Fricas [A]  time = 2.08586, size = 155, normalized size = 4.19 \begin{align*} \frac{2^{\frac{1}{3}} \left (\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}}{\left (3 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i\right )}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^(2/3)*(a+I*a*tan(f*x+e))^(2/3),x, algorithm="fricas")

[Out]

2^(1/3)*(a/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*(3*I*e^(2*I*f*x + 2*I*e) + 3*I
)/f

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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((d*sec(f*x+e))**(2/3)*(a+I*a*tan(f*x+e))**(2/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^(2/3)*(a+I*a*tan(f*x+e))^(2/3),x, algorithm="giac")

[Out]

integrate((d*sec(f*x + e))^(2/3)*(I*a*tan(f*x + e) + a)^(2/3), x)