3.365 \(\int \frac{e^{n \tan ^{-1}(a x)} x^m}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=51 \[ \frac{x^{m+1} F_1\left (m+1;1-\frac{i n}{2},\frac{i n}{2}+1;m+2;i a x,-i a x\right )}{c (m+1)} \]

[Out]

(x^(1 + m)*AppellF1[1 + m, 1 - (I/2)*n, 1 + (I/2)*n, 2 + m, I*a*x, (-I)*a*x])/(c*(1 + m))

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Rubi [A]  time = 0.0923885, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5082, 133} \[ \frac{x^{m+1} F_1\left (m+1;1-\frac{i n}{2},\frac{i n}{2}+1;m+2;i a x,-i a x\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*AppellF1[1 + m, 1 - (I/2)*n, 1 + (I/2)*n, 2 + m, I*a*x, (-I)*a*x])/(c*(1 + m))

Rule 5082

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])

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)} x^m}{c+a^2 c x^2} \, dx &=\frac{\int x^m (1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \, dx}{c}\\ &=\frac{x^{1+m} F_1\left (1+m;1-\frac{i n}{2},1+\frac{i n}{2};2+m;i a x,-i a x\right )}{c (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.152072, size = 96, normalized size = 1.88 \[ \frac{x^m \left (1-e^{2 i \tan ^{-1}(a x)}\right )^{-m} \left (1+e^{2 i \tan ^{-1}(a x)}\right )^m e^{n \tan ^{-1}(a x)} F_1\left (-\frac{i n}{2};m,-m;1-\frac{i n}{2};-e^{2 i \tan ^{-1}(a x)},e^{2 i \tan ^{-1}(a x)}\right )}{a c n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcTan[a*x])*(1 + E^((2*I)*ArcTan[a*x]))^m*x^m*AppellF1[(-I/2)*n, m, -m, 1 - (I/2)*n, -E^((2*I)*ArcTan[a
*x]), E^((2*I)*ArcTan[a*x])])/(a*c*(1 - E^((2*I)*ArcTan[a*x]))^m*n)

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Maple [F]  time = 0.553, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}{x}^{m}}{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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