3.146 \(\int e^{\frac {1}{2} i \tan ^{-1}(a x)} x^m \, dx\)

Optimal. Leaf size=36 \[ \frac {x^{m+1} F_1\left (m+1;\frac {1}{4},-\frac {1}{4};m+2;i a x,-i a x\right )}{m+1} \]

[Out]

x^(1+m)*AppellF1(1+m,-1/4,1/4,2+m,-I*a*x,I*a*x)/(1+m)

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Rubi [A]  time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5062, 133} \[ \frac {x^{m+1} F_1\left (m+1;\frac {1}{4},-\frac {1}{4};m+2;i a x,-i a x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[E^((I/2)*ArcTan[a*x])*x^m,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, 1/4, -1/4, 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^{\frac {1}{2} i \tan ^{-1}(a x)} x^m \, dx &=\int \frac {x^m \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;\frac {1}{4},-\frac {1}{4};2+m;i a x,-i a x\right )}{1+m}\\ \end {align*}

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Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int e^{\frac {1}{2} i \tan ^{-1}(a x)} x^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^((I/2)*ArcTan[a*x])*x^m,x]

[Out]

Integrate[E^((I/2)*ArcTan[a*x])*x^m, x]

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm="fricas")

[Out]

integral(x^m*sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming 0=[0,0]index.cc index_m operator + Error: Bad Argument ValueWarning, need to choose a branch for
the root of a polynomial with parameters. This might be wrong.The choice was done assuming 0=[0,0]index.cc in
dex_m operator + Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with p
arameters. This might be wrong.The choice was done assuming 0=[0,0]index.cc index_m operator + Error: Bad Argu
ment ValueWarning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.T
he choice was done assuming 0=[0,0]index.cc index_m operator + Error: Bad Argument ValueDone

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}\, x^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x)

[Out]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm="maxima")

[Out]

integrate(x^m*sqrt((I*a*x + 1)/sqrt(a^2*x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int x^m\,\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/2),x)

[Out]

int(x^m*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/2)*x**m,x)

[Out]

Integral(x**m*sqrt(I*(a*x - I)/sqrt(a**2*x**2 + 1)), x)

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