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+1/2*I*n,1-1/2*I*n,2+m,-I*a*x,I*a*x)/c/(1+m)

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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])

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])

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 = 0.17, 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[(-1/2*I)*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)

________________________________________________________________________________________

fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}, x\right ) \]

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)

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{a^{2} c \,x^{2}+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________