\(\int e^{-2 \arctan (a x)} (c+a^2 c x^2)^2 \, dx\) [302]

Optimal result

Integrand size = 21, antiderivative size = 53 \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=-\frac {\left (\frac {1}{5}-\frac {3 i}{5}\right ) 2^{1+i} c^2 (1-i a x)^{3-i} \operatorname {Hypergeometric2F1}\left (-2-i,3-i,4-i,\frac {1}{2} (1-i a x)\right )}{a} \] Output:

(-1/5+3/5*I)*2^(1+I)*c^2*(1-I*a*x)^(3-I)*hypergeom([3-I, -2-I],[4-I],1/2-1 
/2*I*a*x)/a
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=-\frac {\left (\frac {1}{5}-\frac {3 i}{5}\right ) 2^{1+i} c^2 (1-i a x)^{3-i} \operatorname {Hypergeometric2F1}\left (-2-i,3-i,4-i,\frac {1}{2} (1-i a x)\right )}{a} \] Input:

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

Output:

((-1/5 + (3*I)/5)*2^(1 + I)*c^2*(1 - I*a*x)^(3 - I)*Hypergeometric2F1[-2 - 
 I, 3 - I, 4 - I, (1 - I*a*x)/2])/a
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.

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

\(\Big \downarrow \) 5596

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {\left (\frac {1}{5}-\frac {3 i}{5}\right ) 2^{1+i} c^2 (1-i a x)^{3-i} \operatorname {Hypergeometric2F1}\left (-2-i,3-i,4-i,\frac {1}{2} (1-i a x)\right )}{a}\)

Input:

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

Output:

((-1/5 + (3*I)/5)*2^(1 + I)*c^2*(1 - I*a*x)^(3 - I)*Hypergeometric2F1[-2 - 
 I, 3 - I, 4 - I, (1 - I*a*x)/2])/a
 

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*e^(-2*arctan(a*x)), x)
 

Sympy [F]

\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=c^{2} \left (\int 2 a^{2} x^{2} e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx\right ) \] Input:

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

Output:

c**2*(Integral(2*a**2*x**2*exp(-2*atan(a*x)), x) + Integral(a**4*x**4*exp( 
-2*atan(a*x)), x) + Integral(exp(-2*atan(a*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=c^{2} \left (\left (\int \frac {x^{4}}{e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{4}+2 \left (\int \frac {x^{2}}{e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{2}+\int \frac {1}{e^{2 \mathit {atan} \left (a x \right )}}d x \right ) \] Input:

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

Output:

c**2*(int(x**4/e**(2*atan(a*x)),x)*a**4 + 2*int(x**2/e**(2*atan(a*x)),x)*a 
**2 + int(1/e**(2*atan(a*x)),x))