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
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[E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^2,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
Time = 0.37 (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[E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^2,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
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]))
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])
\[\int {\mathrm e}^{2 \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{2}d x\]
Input:
int(exp(2*arctan(a*x))*(a^2*c*x^2+c)^2,x)
Output:
int(exp(2*arctan(a*x))*(a^2*c*x^2+c)^2,x)
\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(2*arctan(a*x))*(a^2*c*x^2+c)^2,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)
\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=c^{2} \left (\int 2 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx\right ) \] Input:
integrate(exp(2*atan(a*x))*(a**2*c*x**2+c)**2,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))
\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(2*arctan(a*x))*(a^2*c*x^2+c)^2,x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^2*e^(2*arctan(a*x)), x)
\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(2*arctan(a*x))*(a^2*c*x^2+c)^2,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^2*e^(2*arctan(a*x)), x)
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)
\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=c^{2} \left (\int e^{2 \mathit {atan} \left (a x \right )}d x +\left (\int e^{2 \mathit {atan} \left (a x \right )} x^{4}d x \right ) a^{4}+2 \left (\int e^{2 \mathit {atan} \left (a x \right )} x^{2}d x \right ) a^{2}\right ) \] Input:
int(exp(2*atan(a*x))*(a^2*c*x^2+c)^2,x)
Output:
c**2*(int(e**(2*atan(a*x)),x) + int(e**(2*atan(a*x))*x**4,x)*a**4 + 2*int( e**(2*atan(a*x))*x**2,x)*a**2)