3.273 \(\int \frac{1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 i \sqrt [6]{1+i \tan (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{13}{6},\frac{5}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{2 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}} \]

[Out]

(((-3*I)/2)*Hypergeometric2F1[-1/6, 13/6, 5/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(1/6))/(2^(1/6)*a*
f*(d*Sec[e + f*x])^(1/3))

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Rubi [A]  time = 0.183121, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3505, 3523, 70, 69} \[ -\frac{3 i \sqrt [6]{1+i \tan (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{13}{6},\frac{5}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{2 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/2)*Hypergeometric2F1[-1/6, 13/6, 5/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(1/6))/(2^(1/6)*a*
f*(d*Sec[e + f*x])^(1/3))

Rule 3505

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

Rule 3523

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

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx &=\frac{\left (\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \int \frac{1}{\sqrt [6]{a-i a \tan (e+f x)} (a+i a \tan (e+f x))^{7/6}} \, dx}{\sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\left (a^2 \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-i a x)^{7/6} (a+i a x)^{13/6}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\left (\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{\frac{a+i a \tan (e+f x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{1}{2}+\frac{i x}{2}\right )^{13/6} (a-i a x)^{7/6}} \, dx,x,\tan (e+f x)\right )}{4 \sqrt [6]{2} f \sqrt [3]{d \sec (e+f x)}}\\ &=-\frac{3 i \, _2F_1\left (-\frac{1}{6},\frac{13}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{2 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.90771, size = 112, normalized size = 1.58 \[ \frac{3 (\tan (e+f x)+i) \left (5 (4 i \sin (2 (e+f x))+5 \cos (2 (e+f x))+5)-8 e^{2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{11}{6},-e^{2 i (e+f x)}\right )\right )}{70 a f \sqrt [3]{d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-8*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*
x))] + 5*(5 + 5*Cos[2*(e + f*x)] + (4*I)*Sin[2*(e + f*x)]))*(I + Tan[e + f*x]))/(70*a*f*(d*Sec[e + f*x])^(1/3)
)

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Maple [F]  time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+ia\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [3]{d\sec \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*sec(f*x+e))^(1/3)/(a+I*a*tan(f*x+e)),x)

[Out]

int(1/(d*sec(f*x+e))^(1/3)/(a+I*a*tan(f*x+e)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))^(1/3)/(a+I*a*tan(f*x+e)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2^{\frac{2}{3}} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}}{\left (-21 i \, e^{\left (5 i \, f x + 5 i \, e\right )} - 27 i \, e^{\left (4 i \, f x + 4 i \, e\right )} - 18 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 30 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, e^{\left (i \, f x + i \, e\right )} - 3 i\right )} e^{\left (\frac{2}{3} i \, f x + \frac{2}{3} i \, e\right )} + 28 \,{\left (a d f e^{\left (4 i \, f x + 4 i \, e\right )} - a d f e^{\left (3 i \, f x + 3 i \, e\right )}\right )}{\rm integral}\left (\frac{2^{\frac{2}{3}} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}}{\left (-8 i \, e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, e^{\left (i \, f x + i \, e\right )} - 8 i\right )} e^{\left (\frac{2}{3} i \, f x + \frac{2}{3} i \, e\right )}}{7 \,{\left (a d f e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, a d f e^{\left (2 i \, f x + 2 i \, e\right )} + a d f e^{\left (i \, f x + i \, e\right )}\right )}}, x\right )}{28 \,{\left (a d f e^{\left (4 i \, f x + 4 i \, e\right )} - a d f e^{\left (3 i \, f x + 3 i \, e\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))^(1/3)/(a+I*a*tan(f*x+e)),x, algorithm="fricas")

[Out]

1/28*(2^(2/3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*(-21*I*e^(5*I*f*x + 5*I*e) - 27*I*e^(4*I*f*x + 4*I*e) - 18*I
*e^(3*I*f*x + 3*I*e) - 30*I*e^(2*I*f*x + 2*I*e) + 3*I*e^(I*f*x + I*e) - 3*I)*e^(2/3*I*f*x + 2/3*I*e) + 28*(a*d
*f*e^(4*I*f*x + 4*I*e) - a*d*f*e^(3*I*f*x + 3*I*e))*integral(1/7*2^(2/3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*(
-8*I*e^(2*I*f*x + 2*I*e) - 8*I*e^(I*f*x + I*e) - 8*I)*e^(2/3*I*f*x + 2/3*I*e)/(a*d*f*e^(3*I*f*x + 3*I*e) - 2*a
*d*f*e^(2*I*f*x + 2*I*e) + a*d*f*e^(I*f*x + I*e)), x))/(a*d*f*e^(4*I*f*x + 4*I*e) - a*d*f*e^(3*I*f*x + 3*I*e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))**(1/3)/(a+I*a*tan(f*x+e)),x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}{\left (i \, a \tan \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))^(1/3)/(a+I*a*tan(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*sec(f*x + e))^(1/3)*(I*a*tan(f*x + e) + a)), x)