3.448 \(\int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx\)

Optimal. Leaf size=122 \[ \frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{7 f}+\frac{3 i a (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{7 f} \]

[Out]

(((54*I)/7)*a^3*(d*Sec[e + f*x])^(2/3))/(f*(a + I*a*Tan[e + f*x])^(1/3)) + (((9*I)/7)*a^2*(d*Sec[e + f*x])^(2/
3)*(a + I*a*Tan[e + f*x])^(2/3))/f + (((3*I)/7)*a*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(5/3))/f

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Rubi [A]  time = 0.23504, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3494, 3493} \[ \frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{7 f}+\frac{3 i a (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{7 f} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(8/3),x]

[Out]

(((54*I)/7)*a^3*(d*Sec[e + f*x])^(2/3))/(f*(a + I*a*Tan[e + f*x])^(1/3)) + (((9*I)/7)*a^2*(d*Sec[e + f*x])^(2/
3)*(a + I*a*Tan[e + f*x])^(2/3))/f + (((3*I)/7)*a*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(5/3))/f

Rule 3494

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

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))^{8/3} \, dx &=\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}+\frac{1}{7} (12 a) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3} \, dx\\ &=\frac{9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}+\frac{1}{7} \left (18 a^2\right ) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx\\ &=\frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}\\ \end{align*}

Mathematica [A]  time = 0.595814, size = 100, normalized size = 0.82 \[ \frac{3 a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{5/3} (\sin (e-f x)+i \cos (e-f x)) (5 i \sin (2 (e+f x))+23 \cos (2 (e+f x))+21)}{14 d f (\cos (f x)+i \sin (f x))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(8/3),x]

[Out]

(3*a^2*(d*Sec[e + f*x])^(5/3)*(I*Cos[e - f*x] + Sin[e - f*x])*(21 + 23*Cos[2*(e + f*x)] + (5*I)*Sin[2*(e + f*x
)])*(a + I*a*Tan[e + f*x])^(2/3))/(14*d*f*(Cos[f*x] + I*Sin[f*x])^2)

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Maple [F]  time = 0.14, 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{8}{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))^(8/3),x)

[Out]

int((d*sec(f*x+e))^(2/3)*(a+I*a*tan(f*x+e))^(8/3),x)

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Maxima [B]  time = 2.20946, size = 543, normalized size = 4.45 \begin{align*} \frac{42 \,{\left (-i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (\frac{4}{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}} a^{2} \sin \left (\frac{4}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} \sqrt{\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} a^{\frac{2}{3}} d^{\frac{2}{3}} -{\left (-12 i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (\frac{7}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 12 \cdot 2^{\frac{1}{3}} a^{2} \sin \left (\frac{7}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) +{\left (-84 i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} - 84 i \cdot 2^{\frac{1}{3}} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 168 i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - 84 i \cdot 2^{\frac{1}{3}} a^{2}\right )} \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 ) - 84 \,{\left (2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + 2^{\frac{1}{3}} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \cdot 2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2^{\frac{1}{3}} a^{2}\right )} \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}}}{7 \,{\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{7}{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))^(8/3),x, algorithm="maxima")

[Out]

1/7*(42*(-I*2^(1/3)*a^2*cos(4/3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - 2^(1/3)*a^2*sin(4/3*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)))*sqrt(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) +
1)*a^(2/3)*d^(2/3) - (-12*I*2^(1/3)*a^2*cos(7/3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - 12*2^(1/3)
*a^2*sin(7/3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) + (-84*I*2^(1/3)*a^2*cos(2*f*x + 2*e)^2 - 84*I*2
^(1/3)*a^2*sin(2*f*x + 2*e)^2 - 168*I*2^(1/3)*a^2*cos(2*f*x + 2*e) - 84*I*2^(1/3)*a^2)*cos(1/3*arctan2(sin(2*f
*x + 2*e), cos(2*f*x + 2*e) + 1)) - 84*(2^(1/3)*a^2*cos(2*f*x + 2*e)^2 + 2^(1/3)*a^2*sin(2*f*x + 2*e)^2 + 2*2^
(1/3)*a^2*cos(2*f*x + 2*e) + 2^(1/3)*a^2)*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)^(7/6)*f)

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Fricas [A]  time = 2.06202, size = 305, normalized size = 2.5 \begin{align*} \frac{2 \cdot 2^{\frac{1}{3}}{\left (42 i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 63 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 27 i \, a^{2}\right )} \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}} e^{\left (2 i \, f x + 2 i \, e\right )}}{7 \,{\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + f e^{\left (2 i \, f x + 2 i \, e\right )}\right )}} \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))^(8/3),x, algorithm="fricas")

[Out]

2/7*2^(1/3)*(42*I*a^2*e^(4*I*f*x + 4*I*e) + 63*I*a^2*e^(2*I*f*x + 2*I*e) + 27*I*a^2)*(a/(e^(2*I*f*x + 2*I*e) +
1))^(2/3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*e^(2*I*f*x + 2*I*e)/(f*e^(4*I*f*x + 4*I*e) + f*e^(2*I*f*x + 2*I
*e))

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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))**(8/3),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{8}{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))^(8/3),x, algorithm="giac")

[Out]

integrate((d*sec(f*x + e))^(2/3)*(I*a*tan(f*x + e) + a)^(8/3), x)