3.4.42 \(\int \frac {e^{n \text {ArcTan}(a x)} x}{c+a^2 c x^2} \, dx\) [342]

Optimal. Leaf size=122 \[ \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};1+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a^2 c n} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5190, 80, 71} \begin {gather*} \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;\frac {1}{2} (1-i a x)\right )}{a^2 c n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(n*ArcTan[a*x])*x)/(c + a^2*c*x^2),x]

[Out]

(I*(1 - I*a*x)^((I/2)*n))/(a^2*c*n*(1 + I*a*x)^((I/2)*n)) - (I*2^(1 - (I/2)*n)*(1 - I*a*x)^((I/2)*n)*Hypergeom
etric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a*x)/2])/(a^2*c*n)

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c
, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rule 5190

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 104, normalized size = 0.85 \begin {gather*} -\frac {i e^{n \text {ArcTan}(a x)} \left (\frac {e^{2 i \text {ArcTan}(a x)} \, _2F_1\left (1,1-\frac {i n}{2};2-\frac {i n}{2};-e^{2 i \text {ArcTan}(a x)}\right )}{2 i+n}-\frac {\, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};-e^{2 i \text {ArcTan}(a x)}\right )}{n}\right )}{a^2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*E^(n*ArcTan[a*x])*((E^((2*I)*ArcTan[a*x])*Hypergeometric2F1[1, 1 - (I/2)*n, 2 - (I/2)*n, -E^((2*I)*ArcTa
n[a*x])])/(2*I + n) - Hypergeometric2F1[1, (-1/2*I)*n, 1 - (I/2)*n, -E^((2*I)*ArcTan[a*x])]/n))/(a^2*c)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x}{a^{2} c \,x^{2}+c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*exp(n*atan(a*x))/(a**2*x**2 + 1), x)/c

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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