\(\int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx\) [273]

Optimal result

Integrand size = 28, antiderivative size = 71 \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx=-\frac {3 i \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {13}{6},\frac {5}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{2 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}} \]

[Out]

-3/4*I*hypergeom([-1/6, 13/6],[5/6],1/2-1/2*I*tan(f*x+e))*(1+I*tan(f*x+e))^(1/6)*2^(5/6)/a/f/(d*sec(f*x+e))^(1
/3)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3586, 3604, 72, 71} \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx=-\frac {3 i \sqrt [6]{1+i \tan (e+f x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {13}{6},\frac {5}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{2 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}} \]

[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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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]))

Rule 72

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 3586

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 3604

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \operatorname {Hypergeometric2F1}\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] (verified)

Time = 1.73 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx=\frac {3 \left (-8 e^{2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {11}{6},-e^{2 i (e+f x)}\right )+5 (5+5 \cos (2 (e+f x))+4 i \sin (2 (e+f x)))\right ) (i+\tan (e+f x))}{70 a f \sqrt [3]{d \sec (e+f x)}} \]

[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)
)

Maple [F]

\[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +i a \tan \left (f x +e \right )\right )}d x\]

[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)

Fricas [F]

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

[In]

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

[Out]

-1/28*(3*2^(2/3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*(7*I*e^(5*I*f*x + 5*I*e) + 9*I*e^(4*I*f*x + 4*I*e) + 6*I*
e^(3*I*f*x + 3*I*e) + 10*I*e^(2*I*f*x + 2*I*e) - I*e^(I*f*x + I*e) + 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(-8/7*2^(2/3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(2/3)*(I*e^
(2*I*f*x + 2*I*e) + I*e^(I*f*x + I*e) + 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))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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