\(\int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx\) [262]

Optimal result

Integrand size = 28, antiderivative size = 81 \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {4}{3}}(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}} \] Output:

3/4*AppellF1(4/3,3/2,1,7/3,-I*tan(d*x+c),I*tan(d*x+c))*(1+I*tan(d*x+c))^(1 
/2)*tan(d*x+c)^(4/3)/d/(a+I*a*tan(d*x+c))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [F]

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

Integrate[Tan[c + d*x]^(1/3)/Sqrt[a + I*a*Tan[c + d*x]],x]
 

Output:

Integrate[Tan[c + d*x]^(1/3)/Sqrt[a + I*a*Tan[c + d*x]], x]
 

Rubi [A] (warning: unable to verify)

Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4047, 25, 27, 148, 27, 1013, 1012}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}dx\)

\(\Big \downarrow \) 4047

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 148

\(\displaystyle \frac {3 a^2 \int -\frac {i a^2 \tan ^3(c+d x)}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (\tan ^3(c+d x) a^4+a\right )^{3/2}}d\sqrt [3]{\tan (c+d x)}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 a \int -\frac {i a^3 \tan ^3(c+d x)}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (\tan ^3(c+d x) a^4+a\right )^{3/2}}d\sqrt [3]{\tan (c+d x)}}{d}\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {3 \sqrt {a^3 \tan ^3(c+d x)+1} \int -\frac {i a^3 \tan ^3(c+d x)}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (a^3 \tan ^3(c+d x)+1\right )^{3/2}}d\sqrt [3]{\tan (c+d x)}}{d \sqrt {a^4 \tan ^3(c+d x)+a}}\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {3 a^4 \tan ^4(c+d x) \sqrt {a^3 \tan ^3(c+d x)+1} \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {3}{2},\frac {7}{3},a^3 \tan ^3(c+d x),-a^3 \tan ^3(c+d x)\right )}{4 d \sqrt {a^4 \tan ^3(c+d x)+a}}\)

Input:

Int[Tan[c + d*x]^(1/3)/Sqrt[a + I*a*Tan[c + d*x]],x]
 

Output:

(3*a^4*AppellF1[4/3, 1, 3/2, 7/3, a^3*Tan[c + d*x]^3, -(a^3*Tan[c + d*x]^3 
)]*Tan[c + d*x]^4*Sqrt[1 + a^3*Tan[c + d*x]^3])/(4*d*Sqrt[a + a^4*Tan[c + 
d*x]^3])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 148

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), 
x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1) - 1)*(c 
 + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, 
 d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
 

rule 1012

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] /; 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]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^IntPart[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4047

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(tan(c + d*x)**(1/3)/sqrt(I*a*(tan(c + d*x) - I)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionUnable to transpose Error: Bad Argument ValueDone
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 \sqrt {a}\, i \left (-3 \tan \left (d x +c \right )^{\frac {1}{3}} \sqrt {\tan \left (d x +c \right ) i +1}-5 \left (\int \frac {\tan \left (d x +c \right )^{\frac {4}{3}} \sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{2}+1}d x \right ) \tan \left (d x +c \right )^{2} d -5 \left (\int \frac {\tan \left (d x +c \right )^{\frac {4}{3}} \sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{2}+1}d x \right ) d +\left (\int \frac {\tan \left (d x +c \right )^{\frac {1}{3}} \sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{3}+\tan \left (d x +c \right )}d x \right ) \tan \left (d x +c \right )^{2} d +\left (\int \frac {\tan \left (d x +c \right )^{\frac {1}{3}} \sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{3}+\tan \left (d x +c \right )}d x \right ) d \right )}{3 a d \left (\tan \left (d x +c \right )^{2}+1\right )} \] Input:

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

Output:

(2*sqrt(a)*i*( - 3*tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1) - 5*int((t 
an(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x))/(tan(c + d*x)**2 
 + 1),x)*tan(c + d*x)**2*d - 5*int((tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)* 
i + 1)*tan(c + d*x))/(tan(c + d*x)**2 + 1),x)*d + int((tan(c + d*x)**(1/3) 
*sqrt(tan(c + d*x)*i + 1))/(tan(c + d*x)**3 + tan(c + d*x)),x)*tan(c + d*x 
)**2*d + int((tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1))/(tan(c + d*x)* 
*3 + tan(c + d*x)),x)*d))/(3*a*d*(tan(c + d*x)**2 + 1))