\(\int (c (d \tan (e+f x))^p)^n (a+i a \tan (e+f x))^3 \, dx\) [1334]

Optimal result

Integrand size = 30, antiderivative size = 132 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^3 \, dx=-\frac {3 a^3 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {4 a^3 \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}-\frac {i a^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)} \] Output:

-3*a^3*tan(f*x+e)*(c*(d*tan(f*x+e))^p)^n/f/(n*p+1)+4*a^3*hypergeom([1, n*p 
+1],[n*p+2],I*tan(f*x+e))*tan(f*x+e)*(c*(d*tan(f*x+e))^p)^n/f/(n*p+1)-I*a^ 
3*tan(f*x+e)^2*(c*(d*tan(f*x+e))^p)^n/f/(n*p+2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^3 \, dx=\frac {a^3 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n (-3 (2+n p)+4 (2+n p) \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x))-i (1+n p) \tan (e+f x))}{f (1+n p) (2+n p)} \] Input:

Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3,x]
 

Output:

(a^3*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(-3*(2 + n*p) + 4*(2 + n*p)*Hyp 
ergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]] - I*(1 + n*p)*Tan[e + 
f*x]))/(f*(1 + n*p)*(2 + n*p))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3042, 4853, 27, 2003, 2042, 99, 2009}

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.

\(\displaystyle \int (a+i a \tan (e+f x))^3 \left (c (d \tan (e+f x))^p\right )^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x))^3 \left (c (d \tan (e+f x))^p\right )^ndx\)

\(\Big \downarrow \) 4853

\(\displaystyle \frac {\int \frac {a^3 (i \tan (e+f x)+1)^3 \left (c (d \tan (e+f x))^p\right )^n}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^3 \int \frac {(i \tan (e+f x)+1)^3 \left (c (d \tan (e+f x))^p\right )^n}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 2003

\(\displaystyle \frac {a^3 \int \frac {(i \tan (e+f x)+1)^2 \left (c (d \tan (e+f x))^p\right )^n}{1-i \tan (e+f x)}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 2042

\(\displaystyle \frac {a^3 \tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n \int \frac {(i \tan (e+f x)+1)^2 \tan ^{n p}(e+f x)}{1-i \tan (e+f x)}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {a^3 \tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n \int \left (\frac {4 \tan ^{n p}(e+f x)}{1-i \tan (e+f x)}-3 \tan ^{n p}(e+f x)-i \tan ^{n p+1}(e+f x)\right )d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 \tan ^{-n p}(e+f x) \left (\frac {4 \tan ^{n p+1}(e+f x) \operatorname {Hypergeometric2F1}(1,n p+1,n p+2,i \tan (e+f x))}{n p+1}-\frac {3 \tan ^{n p+1}(e+f x)}{n p+1}-\frac {i \tan ^{n p+2}(e+f x)}{n p+2}\right ) \left (c (d \tan (e+f x))^p\right )^n}{f}\)

Input:

Int[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3,x]
 

Output:

(a^3*(c*(d*Tan[e + f*x])^p)^n*((-3*Tan[e + f*x]^(1 + n*p))/(1 + n*p) + (4* 
Hypergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]]*Tan[e + f*x]^(1 + n 
*p))/(1 + n*p) - (I*Tan[e + f*x]^(2 + n*p))/(2 + n*p)))/(f*Tan[e + f*x]^(n 
*p))
 

Defintions of rubi rules used

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2003

Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : 
> Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} 
, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && 
  !IntegerQ[n]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2042

Int[(u_.)*((c_.)*((d_)*((a_.) + (b_.)*(x_)))^(q_))^(p_), x_Symbol] :> Simp[ 
(c*(d*(a + b*x))^q)^p/(a + b*x)^(p*q)   Int[u*(a + b*x)^(p*q), x], x] /; Fr 
eeQ[{a, b, c, d, q, p}, x] &&  !IntegerQ[q] &&  !IntegerQ[p]
 

rule 3042

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

rule 4853

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa 
ctors[Tan[v], x]}, d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d]], x] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x 
]]
 

Maple [F]

\[\int \left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{3}d x\]

Input:

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^3,x)
 

Output:

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^3,x)
 

Fricas [F]

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

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^3,x, algorithm="fricas 
")
 

Output:

integral(8*a^3*e^(n*p*log((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2 
*I*e) + 1)) + 6*I*f*x + n*log(c) + 6*I*e)/(e^(6*I*f*x + 6*I*e) + 3*e^(4*I* 
f*x + 4*I*e) + 3*e^(2*I*f*x + 2*I*e) + 1), x)
 

Sympy [F]

\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^3 \, dx=- i a^{3} \left (\int i \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int \left (- 3 \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx + \int \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 3 i \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \] Input:

integrate((c*(d*tan(f*x+e))**p)**n*(a+I*a*tan(f*x+e))**3,x)
 

Output:

-I*a**3*(Integral(I*(c*(d*tan(e + f*x))**p)**n, x) + Integral(-3*(c*(d*tan 
(e + f*x))**p)**n*tan(e + f*x), x) + Integral((c*(d*tan(e + f*x))**p)**n*t 
an(e + f*x)**3, x) + Integral(-3*I*(c*(d*tan(e + f*x))**p)**n*tan(e + f*x) 
**2, x))
 

Maxima [F]

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

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^3,x, algorithm="maxima 
")
 

Output:

integrate((I*a*tan(f*x + e) + a)^3*((d*tan(f*x + e))^p*c)^n, x)
 

Giac [F]

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

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((I*a*tan(f*x + e) + a)^3*((d*tan(f*x + e))^p*c)^n, x)
 

Mupad [F(-1)]

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

int((c*(d*tan(e + f*x))^p)^n*(a + a*tan(e + f*x)*1i)^3,x)
 

Output:

int((c*(d*tan(e + f*x))^p)^n*(a + a*tan(e + f*x)*1i)^3, x)
 

Reduce [F]

\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^3 \, dx=\frac {d^{n p} c^{n} a^{3} \left (-\tan \left (f x +e \right )^{n p} \tan \left (f x +e \right )^{2} i n p +4 \tan \left (f x +e \right )^{n p} i n p +8 \tan \left (f x +e \right )^{n p} i +\left (\int \tan \left (f x +e \right )^{n p}d x \right ) f \,n^{2} p^{2}+2 \left (\int \tan \left (f x +e \right )^{n p}d x \right ) f n p -4 \left (\int \frac {\tan \left (f x +e \right )^{n p}}{\tan \left (f x +e \right )}d x \right ) f i \,n^{2} p^{2}-8 \left (\int \frac {\tan \left (f x +e \right )^{n p}}{\tan \left (f x +e \right )}d x \right ) f i n p -3 \left (\int \tan \left (f x +e \right )^{n p} \tan \left (f x +e \right )^{2}d x \right ) f \,n^{2} p^{2}-6 \left (\int \tan \left (f x +e \right )^{n p} \tan \left (f x +e \right )^{2}d x \right ) f n p \right )}{f n p \left (n p +2\right )} \] Input:

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^3,x)
 

Output:

(d**(n*p)*c**n*a**3*( - tan(e + f*x)**(n*p)*tan(e + f*x)**2*i*n*p + 4*tan( 
e + f*x)**(n*p)*i*n*p + 8*tan(e + f*x)**(n*p)*i + int(tan(e + f*x)**(n*p), 
x)*f*n**2*p**2 + 2*int(tan(e + f*x)**(n*p),x)*f*n*p - 4*int(tan(e + f*x)** 
(n*p)/tan(e + f*x),x)*f*i*n**2*p**2 - 8*int(tan(e + f*x)**(n*p)/tan(e + f* 
x),x)*f*i*n*p - 3*int(tan(e + f*x)**(n*p)*tan(e + f*x)**2,x)*f*n**2*p**2 - 
 6*int(tan(e + f*x)**(n*p)*tan(e + f*x)**2,x)*f*n*p))/(f*n*p*(n*p + 2))