\(\int \tan ^3(c+d x) (a+i a \tan (c+d x))^m \, dx\) [326]

Optimal result

Integrand size = 24, antiderivative size = 144 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^m \, dx=-\frac {2 (a+i a \tan (c+d x))^m}{d m (2+m)}+\frac {\operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}+\frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (2+m)}-\frac {m (a+i a \tan (c+d x))^{1+m}}{a d \left (2+3 m+m^2\right )} \] Output:

-2*(a+I*a*tan(d*x+c))^m/d/m/(2+m)+1/2*hypergeom([1, m],[1+m],1/2+1/2*I*tan 
(d*x+c))*(a+I*a*tan(d*x+c))^m/d/m+tan(d*x+c)^2*(a+I*a*tan(d*x+c))^m/d/(2+m 
)-m*(a+I*a*tan(d*x+c))^(1+m)/a/d/(m^2+3*m+2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^m \, dx=\frac {(a+i a \tan (c+d x))^m \left (-2 \left (2+2 m+m^2\right )+\left (2+3 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (c+d x))\right )-2 i m^2 \tan (c+d x)+2 m (1+m) \tan ^2(c+d x)\right )}{2 d m (1+m) (2+m)} \] Input:

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

Output:

((a + I*a*Tan[c + d*x])^m*(-2*(2 + 2*m + m^2) + (2 + 3*m + m^2)*Hypergeome 
tric2F1[1, m, 1 + m, (1 + I*Tan[c + d*x])/2] - (2*I)*m^2*Tan[c + d*x] + 2* 
m*(1 + m)*Tan[c + d*x]^2))/(2*d*m*(1 + m)*(2 + m))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 4043, 3042, 4075, 3042, 4010, 3042, 3962, 78}

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.

\(\displaystyle \int \tan ^3(c+d x) (a+i a \tan (c+d x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (c+d x)^3 (a+i a \tan (c+d x))^mdx\)

\(\Big \downarrow \) 4043

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4075

\(\displaystyle \frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac {\int (i \tan (c+d x) a+a)^m (2 a \tan (c+d x)-i a m)dx+\frac {m (a+i a \tan (c+d x))^{m+1}}{d (m+1)}}{a (m+2)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac {\int (i \tan (c+d x) a+a)^m (2 a \tan (c+d x)-i a m)dx+\frac {m (a+i a \tan (c+d x))^{m+1}}{d (m+1)}}{a (m+2)}\)

\(\Big \downarrow \) 4010

\(\displaystyle \frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac {-i a (m+2) \int (i \tan (c+d x) a+a)^mdx+\frac {2 a (a+i a \tan (c+d x))^m}{d m}+\frac {m (a+i a \tan (c+d x))^{m+1}}{d (m+1)}}{a (m+2)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac {-i a (m+2) \int (i \tan (c+d x) a+a)^mdx+\frac {2 a (a+i a \tan (c+d x))^m}{d m}+\frac {m (a+i a \tan (c+d x))^{m+1}}{d (m+1)}}{a (m+2)}\)

\(\Big \downarrow \) 3962

\(\displaystyle \frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac {-\frac {a^2 (m+2) \int \frac {(i \tan (c+d x) a+a)^{m-1}}{a-i a \tan (c+d x)}d(i a \tan (c+d x))}{d}+\frac {2 a (a+i a \tan (c+d x))^m}{d m}+\frac {m (a+i a \tan (c+d x))^{m+1}}{d (m+1)}}{a (m+2)}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac {-\frac {a (m+2) (a+i a \tan (c+d x))^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {i \tan (c+d x) a+a}{2 a}\right )}{2 d m}+\frac {2 a (a+i a \tan (c+d x))^m}{d m}+\frac {m (a+i a \tan (c+d x))^{m+1}}{d (m+1)}}{a (m+2)}\)

Input:

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

Output:

(Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m)/(d*(2 + m)) - ((2*a*(a + I*a*Tan 
[c + d*x])^m)/(d*m) - (a*(2 + m)*Hypergeometric2F1[1, m, 1 + m, (a + I*a*T 
an[c + d*x])/(2*a)]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) + (m*(a + I*a*Tan[c 
+ d*x])^(1 + m))/(d*(1 + m)))/(a*(2 + m))
 

Defintions of rubi rules used

rule 78

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

rule 3042

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

rule 3962

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d   S 
ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b 
, c, d, n}, x] && EqQ[a^2 + b^2, 0]
 

rule 4010

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

rule 4043

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

rule 4075

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B 
*d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f* 
x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, 
d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((2*a*e^(2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^m*(I*e^(6*I*d 
*x + 6*I*c) - 3*I*e^(4*I*d*x + 4*I*c) + 3*I*e^(2*I*d*x + 2*I*c) - I)/(e^(6 
*I*d*x + 6*I*c) + 3*e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c) + 1), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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