\(\int (c (d \tan (e+f x))^p)^n (a+i a \tan (e+f x))^2 \, dx\) [1335]

Optimal result

Integrand size = 30, antiderivative size = 93 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=-\frac {a^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {2 a^2 \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \] Output:

-a^2*tan(f*x+e)*(c*(d*tan(f*x+e))^p)^n/f/(n*p+1)+2*a^2*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)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.62 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=\frac {a^2 (-1+2 \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x))) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f+f n p} \] Input:

Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04, 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, 90, 74}

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.

\(\displaystyle \int (a+i a \tan (e+f x))^2 \left (c (d \tan (e+f x))^p\right )^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x))^2 \left (c (d \tan (e+f x))^p\right )^ndx\)

\(\Big \downarrow \) 4853

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2003

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

\(\Big \downarrow \) 2042

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 74

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

Input:

Int[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^2,x]
 

Output:

(a^2*(c*(d*Tan[e + f*x])^p)^n*(-(Tan[e + f*x]^(1 + n*p)/(1 + n*p)) + (2*Hy 
pergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]]*Tan[e + f*x]^(1 + n*p 
))/(1 + 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 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

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 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]

\[\int \left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{2}d x\]

Input:

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x)
 

Output:

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x)
 

Fricas [F]

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

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x, algorithm="fricas 
")
 

Output:

integral(4*a^2*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)) + 4*I*f*x + n*log(c) + 4*I*e)/(e^(4*I*f*x + 4*I*e) + 2*e^(2*I* 
f*x + 2*I*e) + 1), x)
 

Sympy [F]

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

integrate((c*(d*tan(f*x+e))**p)**n*(a+I*a*tan(f*x+e))**2,x)
 

Output:

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

Maxima [F]

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

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x, algorithm="maxima 
")
 

Output:

integrate((I*a*tan(f*x + e) + a)^2*((d*tan(f*x + e))^p*c)^n, x)
 

Giac [F]

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

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((I*a*tan(f*x + e) + a)^2*((d*tan(f*x + e))^p*c)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=\int {\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \] Input:

int((c*(d*tan(e + f*x))^p)^n*(a + a*tan(e + f*x)*1i)^2,x)
 

Output:

int((c*(d*tan(e + f*x))^p)^n*(a + a*tan(e + f*x)*1i)^2, x)
 

Reduce [F]

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

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x)
 

Output:

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