\(\int \frac {\sqrt {a+i a \tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)} \, dx\) [252]

Optimal result

Integrand size = 28, antiderivative size = 80 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)} \, dx=\frac {3 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \]

[Out]

3*a*AppellF1(1/3,1/2,1,4/3,-I*tan(d*x+c),I*tan(d*x+c))*(1+I*tan(d*x+c))^(1/2)*tan(d*x+c)^(1/3)/d/(a+I*a*tan(d*
x+c))^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 80, 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, 441, 440} \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)} \, dx=\frac {3 a \sqrt {1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-i \tan (c+d x),i \tan (c+d x)\right )}{d \sqrt {a+i a \tan (c+d x)}} \]

[In]

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

[Out]

(3*a*AppellF1[1/3, 1/2, 1, 4/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1/3)
)/(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 440

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

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[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 {1}{\left (-\frac {i x}{a}\right )^{2/3} \sqrt {a+x} \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 {1}{\sqrt {a+i a x^3} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d} \\ & = -\frac {\left (3 a^3 \sqrt {1+i \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i x^3} \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 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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