\(\int e^{\arctan (a x)} (c+a^2 c x^2) \, dx\) [246]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (-1+\frac {i}{2},2+\frac {i}{2},3+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \]

[Out]

(1/17+4/17*I)*2^(2-1/2*I)*c*(1-I*a*x)^(2+1/2*I)*hypergeom([2+1/2*I, -1+1/2*I],[3+1/2*I],1/2-1/2*I*a*x)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5181, 71} \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (-1+\frac {i}{2},2+\frac {i}{2},3+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \]

[In]

Int[E^ArcTan[a*x]*(c + a^2*c*x^2),x]

[Out]

((1/17 + (4*I)/17)*2^(2 - I/2)*c*(1 - I*a*x)^(2 + I/2)*Hypergeometric2F1[-1 + I/2, 2 + I/2, 3 + I/2, (1 - I*a*
x)/2])/a

Rule 71

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 5181

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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])

Rubi steps \begin{align*} \text {integral}& = c \int (1-i a x)^{1+\frac {i}{2}} (1+i a x)^{1-\frac {i}{2}} \, dx \\ & = \frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (-1+\frac {i}{2},2+\frac {i}{2},3+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \operatorname {Hypergeometric2F1}\left (-1+\frac {i}{2},2+\frac {i}{2},3+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a} \]

[In]

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

[Out]

((1/17 + (4*I)/17)*2^(2 - I/2)*c*(1 - I*a*x)^(2 + I/2)*Hypergeometric2F1[-1 + I/2, 2 + I/2, 3 + I/2, (1 - I*a*
x)/2])/a

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(exp(atan(a*x))*(c + a^2*c*x^2),x)

[Out]

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