\(\int e^{-2 \arctan (a x)} (c+a^2 c x^2)^p \, dx\) [301]

Optimal result

Integrand size = 21, antiderivative size = 90 \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{i+p} (1-i a x)^{(1-i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-i-p,(1-i)+p,(2-i)+p,\frac {1}{2} (1-i a x)\right )}{a ((1-i)+p)} \] Output:

I*2^(I+p)*(1-I*a*x)^(1-I+p)*(a^2*c*x^2+c)^p*hypergeom([-I-p, 1-I+p],[2-I+p 
],1/2-1/2*I*a*x)/a/(1-I+p)/((a^2*x^2+1)^p)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{i+p} (1-i a x)^{(1-i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-i-p,(1-i)+p,(2-i)+p,\frac {1}{2} (1-i a x)\right )}{a ((1-i)+p)} \] Input:

Integrate[(c + a^2*c*x^2)^p/E^(2*ArcTan[a*x]),x]
 

Output:

(I*2^(I + p)*(1 - I*a*x)^((1 - I) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1 
[-I - p, (1 - I) + p, (2 - I) + p, (1 - I*a*x)/2])/(a*((1 - I) + p)*(1 + a 
^2*x^2)^p)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5599, 5596, 79}

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.

\(\displaystyle \int e^{-2 \arctan (a x)} \left (a^2 c x^2+c\right )^p \, dx\)

\(\Big \downarrow \) 5599

\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int e^{-2 \arctan (a x)} \left (a^2 x^2+1\right )^pdx\)

\(\Big \downarrow \) 5596

\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int (1-i a x)^{p-i} (i a x+1)^{p+i}dx\)

\(\Big \downarrow \) 79

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

Input:

Int[(c + a^2*c*x^2)^p/E^(2*ArcTan[a*x]),x]
 

Output:

(I*2^(I + p)*(1 - I*a*x)^((1 - I) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1 
[-I - p, (1 - I) + p, (2 - I) + p, (1 - I*a*x)/2])/(a*((1 - I) + p)*(1 + a 
^2*x^2)^p)
 

Defintions of rubi rules used

rule 79

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

rule 5596

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[c^p   Int[(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] 
/; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
 

rule 5599

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p])   Int[ 
(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && E 
qQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

\[\int \left (a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 \arctan \left (a x \right )}d x\]

Input:

int((a^2*c*x^2+c)^p/exp(2*arctan(a*x)),x)
 

Output:

int((a^2*c*x^2+c)^p/exp(2*arctan(a*x)),x)
 

Fricas [F]

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

integrate((a^2*c*x^2+c)^p/exp(2*arctan(a*x)),x, algorithm="fricas")
 

Output:

integral((a^2*c*x^2 + c)^p*e^(-2*arctan(a*x)), x)
 

Sympy [F]

\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx \] Input:

integrate((a**2*c*x**2+c)**p/exp(2*atan(a*x)),x)
 

Output:

Integral((c*(a**2*x**2 + 1))**p*exp(-2*atan(a*x)), x)
 

Maxima [F]

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

integrate((a^2*c*x^2+c)^p/exp(2*arctan(a*x)),x, algorithm="maxima")
 

Output:

integrate((a^2*c*x^2 + c)^p*e^(-2*arctan(a*x)), x)
 

Giac [F]

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

integrate((a^2*c*x^2+c)^p/exp(2*arctan(a*x)),x, algorithm="giac")
 

Output:

integrate((a^2*c*x^2 + c)^p*e^(-2*arctan(a*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int {\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \] Input:

int(exp(-2*atan(a*x))*(c + a^2*c*x^2)^p,x)
 

Output:

int(exp(-2*atan(a*x))*(c + a^2*c*x^2)^p, x)
 

Reduce [F]

\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int \frac {\left (a^{2} c \,x^{2}+c \right )^{p}}{e^{2 \mathit {atan} \left (a x \right )}}d x \] Input:

int((a^2*c*x^2+c)^p/exp(2*atan(a*x)),x)
 

Output:

int((a**2*c*x**2 + c)**p/e**(2*atan(a*x)),x)