Integrand size = 19, antiderivative size = 84 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=-\frac {2^{2-\frac {i n}{2}} c (1-i a x)^{2+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (-1+\frac {i n}{2},2+\frac {i n}{2},3+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a (4 i-n)} \] Output:
-2^(2-1/2*I*n)*c*(1-I*a*x)^(2+1/2*I*n)*hypergeom([-1+1/2*I*n, 2+1/2*I*n],[ 3+1/2*I*n],1/2-1/2*I*a*x)/a/(4*I-n)
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.05 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {i 2^{1-\frac {i n}{2}} c (1-i a x)^{2+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (-1+\frac {i n}{2},2+\frac {i n}{2},3+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a \left (2+\frac {i n}{2}\right )} \] Input:
Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2),x]
Output:
(I*2^(1 - (I/2)*n)*c*(1 - I*a*x)^(2 + (I/2)*n)*Hypergeometric2F1[-1 + (I/2 )*n, 2 + (I/2)*n, 3 + (I/2)*n, (1 - I*a*x)/2])/(a*(2 + (I/2)*n))
Time = 0.41 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int \left (a^2 c x^2+c\right ) e^{n \arctan (a x)} \, dx\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle c \int (1-i a x)^{\frac {i n}{2}+1} (i a x+1)^{1-\frac {i n}{2}}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {c 2^{2-\frac {i n}{2}} (1-i a x)^{2+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}-1,\frac {i n}{2}+2,\frac {i n}{2}+3,\frac {1}{2} (1-i a x)\right )}{a (-n+4 i)}\) |
Input:
Int[E^(n*ArcTan[a*x])*(c + a^2*c*x^2),x]
Output:
-((2^(2 - (I/2)*n)*c*(1 - I*a*x)^(2 + (I/2)*n)*Hypergeometric2F1[-1 + (I/2 )*n, 2 + (I/2)*n, 3 + (I/2)*n, (1 - I*a*x)/2])/(a*(4*I - n)))
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])
\[\int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )d x\]
Input:
int(exp(n*arctan(a*x))*(a^2*c*x^2+c),x)
Output:
int(exp(n*arctan(a*x))*(a^2*c*x^2+c),x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*(a^2*c*x^2+c),x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)*e^(n*arctan(a*x)), x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=c \left (\int a^{2} x^{2} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \] Input:
integrate(exp(n*atan(a*x))*(a**2*c*x**2+c),x)
Output:
c*(Integral(a**2*x**2*exp(n*atan(a*x)), x) + Integral(exp(n*atan(a*x)), x) )
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*(a^2*c*x^2+c),x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)*e^(n*arctan(a*x)), x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*(a^2*c*x^2+c),x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)*e^(n*arctan(a*x)), x)
Timed out. \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \] Input:
int(exp(n*atan(a*x))*(c + a^2*c*x^2),x)
Output:
int(exp(n*atan(a*x))*(c + a^2*c*x^2), x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=c \left (\int e^{\mathit {atan} \left (a x \right ) n}d x +\left (\int e^{\mathit {atan} \left (a x \right ) n} x^{2}d x \right ) a^{2}\right ) \] Input:
int(exp(n*atan(a*x))*(a^2*c*x^2+c),x)
Output:
c*(int(e**(atan(a*x)*n),x) + int(e**(atan(a*x)*n)*x**2,x)*a**2)