3.153 \(\int e^{i n \tan ^{-1}(a x)} x^m \, dx\)

Optimal. Leaf size=40 \[ \frac {x^{m+1} F_1\left (m+1;\frac {n}{2},-\frac {n}{2};m+2;i a x,-i a x\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5062, 133} \[ \frac {x^{m+1} F_1\left (m+1;\frac {n}{2},-\frac {n}{2};m+2;i a x,-i a x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[E^(I*n*ArcTan[a*x])*x^m,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, n/2, -n/2, 2 + m, I*a*x, (-I)*a*x])/(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 5062

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2
), x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps

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

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \[ \int e^{i n \tan ^{-1}(a x)} x^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(I*n*ArcTan[a*x])*x^m,x]

[Out]

Integrate[E^(I*n*ArcTan[a*x])*x^m, x]

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))*x^m,x, algorithm="fricas")

[Out]

integral(x^m/(-(a*x + I)/(a*x - I))^(1/2*n), 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(I*n*arctan(a*x))*x^m,x, algorithm="giac")

[Out]

sage0*x

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maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{i n \arctan \left (a x \right )} x^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(I*n*arctan(a*x))*x^m,x)

[Out]

int(exp(I*n*arctan(a*x))*x^m,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*arctan(a*x))*x^m,x, algorithm="maxima")

[Out]

integrate(x^m*e^(I*n*arctan(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*exp(n*atan(a*x)*1i),x)

[Out]

int(x^m*exp(n*atan(a*x)*1i), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(I*n*atan(a*x))*x**m,x)

[Out]

Integral(x**m*exp(I*n*atan(a*x)), x)

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