3.134 \(\int e^{\frac{1}{4} \tanh ^{-1}(a x)} x^m \, dx\)

Optimal. Leaf size=31 \[ \frac{x^{m+1} F_1\left (m+1;\frac{1}{8},-\frac{1}{8};m+2;a x,-a x\right )}{m+1} \]

[Out]

(x^(1 + m)*AppellF1[1 + m, 1/8, -1/8, 2 + m, a*x, -(a*x)])/(1 + m)

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Rubi [A]  time = 0.0275486, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6126, 133} \[ \frac{x^{m+1} F_1\left (m+1;\frac{1}{8},-\frac{1}{8};m+2;a x,-a x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[E^(ArcTanh[a*x]/4)*x^m,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, 1/8, -1/8, 2 + m, a*x, -(a*x)])/(1 + m)

Rule 6126

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

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 e^{\frac{1}{4} \tanh ^{-1}(a x)} x^m \, dx &=\int \frac{x^m \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx\\ &=\frac{x^{1+m} F_1\left (1+m;\frac{1}{8},-\frac{1}{8};2+m;a x,-a x\right )}{1+m}\\ \end{align*}

Mathematica [F]  time = 0.302039, size = 0, normalized size = 0. \[ \int e^{\frac{1}{4} \tanh ^{-1}(a x)} x^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(ArcTanh[a*x]/4)*x^m,x]

[Out]

Integrate[E^(ArcTanh[a*x]/4)*x^m, x]

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}{x}^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^m,x, algorithm="maxima")

[Out]

integrate(x^m*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^m,x, algorithm="fricas")

[Out]

integral(x^m*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)*x**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^m,x, algorithm="giac")

[Out]

integrate(x^m*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4), x)