Integrand size = 21, antiderivative size = 115 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {2^{1-\frac {i n}{2}+p} (1-i a x)^{1+\frac {i n}{2}+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}-p,1+\frac {i n}{2}+p,2+\frac {i n}{2}+p,\frac {1}{2} (1-i a x)\right )}{a (n-2 i (1+p))} \] Output:
2^(1-1/2*I*n+p)*(1-I*a*x)^(1+1/2*I*n+p)*(a^2*c*x^2+c)^p*hypergeom([1/2*I*n -p, 1+1/2*I*n+p],[2+1/2*I*n+p],1/2-1/2*I*a*x)/a/(n-2*I*(p+1))/((a^2*x^2+1) ^p)
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {2^{1-\frac {i n}{2}+p} (1-i a x)^{1+\frac {i n}{2}+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}-p,1+\frac {i n}{2}+p,2+\frac {i n}{2}+p,\frac {1}{2} (1-i a x)\right )}{a (n-2 i (1+p))} \] Input:
Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^p,x]
Output:
(2^(1 - (I/2)*n + p)*(1 - I*a*x)^(1 + (I/2)*n + p)*(c + a^2*c*x^2)^p*Hyper geometric2F1[(I/2)*n - p, 1 + (I/2)*n + p, 2 + (I/2)*n + p, (1 - I*a*x)/2] )/(a*(n - (2*I)*(1 + p))*(1 + a^2*x^2)^p)
Time = 0.56 (sec) , antiderivative size = 115, 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.
\(\displaystyle \int e^{n \arctan (a x)} \left (a^2 c x^2+c\right )^p \, dx\) |
\(\Big \downarrow \) 5599 |
\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int e^{n \arctan (a x)} \left (a^2 x^2+1\right )^pdx\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int (1-i a x)^{\frac {i n}{2}+p} (i a x+1)^{p-\frac {i n}{2}}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {2^{-\frac {i n}{2}+p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p (1-i a x)^{\frac {i n}{2}+p+1} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}-p,\frac {i n}{2}+p+1,\frac {i n}{2}+p+2,\frac {1}{2} (1-i a x)\right )}{a (n-2 i (p+1))}\) |
Input:
Int[E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^p,x]
Output:
(2^(1 - (I/2)*n + p)*(1 - I*a*x)^(1 + (I/2)*n + p)*(c + a^2*c*x^2)^p*Hyper geometric2F1[(I/2)*n - p, 1 + (I/2)*n + p, 2 + (I/2)*n + p, (1 - I*a*x)/2] )/(a*(n - (2*I)*(1 + p))*(1 + a^2*x^2)^p)
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[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])
\[\int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}d x\]
Input:
int(exp(n*arctan(a*x))*(a^2*c*x^2+c)^p,x)
Output:
int(exp(n*arctan(a*x))*(a^2*c*x^2+c)^p,x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*(a^2*c*x^2+c)^p,x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)^p*e^(n*arctan(a*x)), x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{n \operatorname {atan}{\left (a x \right )}}\, dx \] Input:
integrate(exp(n*atan(a*x))*(a**2*c*x**2+c)**p,x)
Output:
Integral((c*(a**2*x**2 + 1))**p*exp(n*atan(a*x)), x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*(a^2*c*x^2+c)^p,x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^p*e^(n*arctan(a*x)), x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*(a^2*c*x^2+c)^p,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^p*e^(n*arctan(a*x)), x)
Timed out. \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \] Input:
int(exp(n*atan(a*x))*(c + a^2*c*x^2)^p,x)
Output:
int(exp(n*atan(a*x))*(c + a^2*c*x^2)^p, x)
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int e^{\mathit {atan} \left (a x \right ) n} \left (a^{2} c \,x^{2}+c \right )^{p}d x \] Input:
int(exp(n*atan(a*x))*(a^2*c*x^2+c)^p,x)
Output:
int(e**(atan(a*x)*n)*(a**2*c*x**2 + c)**p,x)