\(\int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\) [261]

Optimal result

Integrand size = 28, antiderivative size = 81 \[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},1,\frac {8}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x)}{5 d \sqrt {a+i a \tan (c+d x)}} \]

[Out]

3/5*AppellF1(5/3,3/2,1,8/3,-I*tan(d*x+c),I*tan(d*x+c))*(1+I*tan(d*x+c))^(1/2)*tan(d*x+c)^(5/3)/d/(a+I*a*tan(d*
x+c))^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 81, 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 = {3645, 129, 525, 524} \[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x) \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},1,\frac {8}{3},-i \tan (c+d x),i \tan (c+d x)\right )}{5 d \sqrt {a+i a \tan (c+d x)}} \]

[In]

Int[Tan[c + d*x]^(2/3)/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(3*AppellF1[5/3, 3/2, 1, 8/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(5/3))/
(5*d*Sqrt[a + I*a*Tan[c + d*x]])

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 3645

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^{2/3}}{(a+x)^{3/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {x^4}{\left (a+i a x^3\right )^{3/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d} \\ & = -\frac {\left (3 a^2 \sqrt {1+i \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+i x^3\right )^{3/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {3 \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},1,\frac {8}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x)}{5 d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx \]

[In]

Integrate[Tan[c + d*x]^(2/3)/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

Integrate[Tan[c + d*x]^(2/3)/Sqrt[a + I*a*Tan[c + d*x]], x]

Maple [F]

\[\int \frac {\tan ^{\frac {2}{3}}\left (d x +c \right )}{\sqrt {a +i a \tan \left (d x +c \right )}}d x\]

[In]

int(tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(1/2),x)

[Out]

int(tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(1/2),x)

Fricas [F]

\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-(2*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(
-I*e^(4*I*d*x + 4*I*c) - 2*I*e^(2*I*d*x + 2*I*c) - I) - (a*d*e^(3*I*d*x + 3*I*c) - 4*a*d*e^(2*I*d*x + 2*I*c) +
4*a*d*e^(I*d*x + I*c))*integral(1/6*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*((-I*e^(2*I*d*x + 2*I*c) + I)/(
e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(-3*I*e^(5*I*d*x + 5*I*c) + 66*I*e^(4*I*d*x + 4*I*c) - 32*I*e^(3*I*d*x + 3*I*c
) + 64*I*e^(2*I*d*x + 2*I*c) - 29*I*e^(I*d*x + I*c) - 2*I)/(a*d*e^(5*I*d*x + 5*I*c) - 6*a*d*e^(4*I*d*x + 4*I*c
) + 11*a*d*e^(3*I*d*x + 3*I*c) - 2*a*d*e^(2*I*d*x + 2*I*c) - 12*a*d*e^(I*d*x + I*c) + 8*a*d), x))/(a*d*e^(3*I*
d*x + 3*I*c) - 4*a*d*e^(2*I*d*x + 2*I*c) + 4*a*d*e^(I*d*x + I*c))

Sympy [F]

\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\tan ^{\frac {2}{3}}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]

[In]

integrate(tan(d*x+c)**(2/3)/(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

Integral(tan(c + d*x)**(2/3)/sqrt(I*a*(tan(c + d*x) - I)), x)

Maxima [F]

\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(tan(d*x + c)^(2/3)/sqrt(I*a*tan(d*x + c) + a), x)

Giac [F]

\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(tan(d*x + c)^(2/3)/sqrt(I*a*tan(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]

[In]

int(tan(c + d*x)^(2/3)/(a + a*tan(c + d*x)*1i)^(1/2),x)

[Out]

int(tan(c + d*x)^(2/3)/(a + a*tan(c + d*x)*1i)^(1/2), x)