\(\int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\) [290]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {3 i \operatorname {AppellF1}\left (-\frac {4}{3},-m,1,-\frac {1}{3},1+i \tan (c+d x),\frac {1}{2} (1+i \tan (c+d x))\right ) (-i \tan (c+d x))^{-m} \tan ^m(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}} \] Output:

3/8*I*AppellF1(-4/3,-m,1,-1/3,1+I*tan(d*x+c),1/2+1/2*I*tan(d*x+c))*tan(d*x 
+c)^m/d/((-I*tan(d*x+c))^m)/(a+I*a*tan(d*x+c))^(4/3)
 

Mathematica [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx \] Input:

Integrate[Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(4/3),x]
 

Output:

Integrate[Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(4/3), x]
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 4047, 25, 27, 152, 150}

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.

\(\displaystyle \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^m}{(a+i a \tan (c+d x))^{4/3}}dx\)

\(\Big \downarrow \) 4047

\(\displaystyle \frac {i a^2 \int -\frac {\tan ^m(c+d x)}{a (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/3}}d(i a \tan (c+d x))}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i a^2 \int \frac {\tan ^m(c+d x)}{a (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/3}}d(i a \tan (c+d x))}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {i a \int \frac {\tan ^m(c+d x)}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/3}}d(i a \tan (c+d x))}{d}\)

\(\Big \downarrow \) 152

\(\displaystyle -\frac {i \sqrt [3]{1+i \tan (c+d x)} \int \frac {\tan ^m(c+d x)}{(i \tan (c+d x)+1)^{7/3} (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{a d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 150

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

Input:

Int[Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(4/3),x]
 

Output:

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

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 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 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]

\[\int \frac {\tan \left (d x +c \right )^{m}}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}d x\]

Input:

int(tan(d*x+c)^m/(a+I*a*tan(d*x+c))^(4/3),x)
 

Output:

int(tan(d*x+c)^m/(a+I*a*tan(d*x+c))^(4/3),x)
 

Fricas [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(tan(d*x+c)^m/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="fricas")
 

Output:

integral(1/4*2^(2/3)*((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 
1))^m*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(e^(4*I*d*x + 4*I*c) + 2*e^(2*I* 
d*x + 2*I*c) + 1)*e^(-8/3*I*d*x - 8/3*I*c)/a^2, x)
 

Sympy [F]

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

integrate(tan(d*x+c)**m/(a+I*a*tan(d*x+c))**(4/3),x)
 

Output:

Integral(tan(c + d*x)**m/(I*a*(tan(c + d*x) - I))**(4/3), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(tan(d*x+c)^m/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(tan(d*x+c)^m/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="giac")
 

Output:

integrate(tan(d*x + c)^m/(I*a*tan(d*x + c) + a)^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}} \,d x \] Input:

int(tan(c + d*x)^m/(a + a*tan(c + d*x)*1i)^(4/3),x)
 

Output:

int(tan(c + d*x)^m/(a + a*tan(c + d*x)*1i)^(4/3), x)
 

Reduce [F]

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

int(tan(d*x+c)^m/(a+I*a*tan(d*x+c))^(4/3),x)
                                                                                    
                                                                                    
 

Output:

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