3.1360 \(\int e^{n \tanh ^{-1}(a x)} x^m (c-a^2 c x^2)^p \, dx\)

Optimal. Leaf size=70 \[ \frac{x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac{1}{2} (n-2 p),-\frac{n}{2}-p;m+2;a x,-a x\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.161412, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 133} \[ \frac{x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac{1}{2} (n-2 p),-\frac{n}{2}-p;m+2;a x,-a x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^p,x]

[Out]

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

Rule 6153

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c^IntPart[p]*(c +
d*x^2)^FracPart[p])/(1 - a^2*x^2)^FracPart[p], Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a,
 c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[n/2]

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[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 e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{n \tanh ^{-1}(a x)} x^m \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^m (1-a x)^{-\frac{n}{2}+p} (1+a x)^{\frac{n}{2}+p} \, dx\\ &=\frac{x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (1+m;\frac{1}{2} (n-2 p),-\frac{n}{2}-p;2+m;a x,-a x\right )}{1+m}\\ \end{align*}

Mathematica [F]  time = 0.478661, size = 0, normalized size = 0. \[ \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(n*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^p,x]

[Out]

Integrate[E^(n*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^p, x]

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Maple [F]  time = 0.307, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{m} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^p,x)

[Out]

int(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^p,x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^p*x^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^p,x, algorithm="fricas")

[Out]

integral((-a^2*c*x^2 + c)^p*x^m*((a*x + 1)/(a*x - 1))^(1/2*n), 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(exp(n*atanh(a*x))*x**m*(-a**2*c*x**2+c)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^p,x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^p*x^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)