3.246 \(\int e^{\tan ^{-1}(a x)} (c+a^2 c x^2) \, dx\)

Optimal. Leaf size=61 \[ \frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \, _2F_1\left (-1+\frac {i}{2},2+\frac {i}{2};3+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \]

[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

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5073, 69} \[ \frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \, _2F_1\left (-1+\frac {i}{2},2+\frac {i}{2};3+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \]

Antiderivative was successfully verified.

[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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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 5073

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*} \int e^{\tan ^{-1}(a x)} \left (c+a^2 c x^2\right ) \, dx &=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}} \, _2F_1\left (-1+\frac {i}{2},2+\frac {i}{2};3+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 61, normalized size = 1.00 \[ \frac {\left (\frac {1}{17}+\frac {4 i}{17}\right ) 2^{2-\frac {i}{2}} c (1-i a x)^{2+\frac {i}{2}} \, _2F_1\left (-1+\frac {i}{2},2+\frac {i}{2};3+\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \]

Antiderivative was successfully verified.

[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

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} e^{\left (\arctan \left (a x\right )\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )} e^{\left (\arctan \left (a x\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}\, dx + \int e^{\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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