3.14.56 \(\int e^{n \tanh ^{-1}(a x)} x^m (c-a^2 c x^2)^2 \, dx\) [1356]

Optimal. Leaf size=42 \[ \frac {c^2 x^{1+m} F_1\left (1+m;\frac {1}{2} (-4+n),-2-\frac {n}{2};2+m;a x,-a x\right )}{1+m} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 138} \begin {gather*} \frac {c^2 x^{m+1} F_1\left (m+1;\frac {n-4}{2},-\frac {n}{2}-2;m+2;a x,-a x\right )}{m+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^(1 + m)*AppellF1[1 + m, (-4 + n)/2, -2 - n/2, 2 + m, a*x, -(a*x)])/(1 + m)

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 6285

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])

Rubi steps

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

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Mathematica [F]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} x^{m} \left (-a^{2} c \,x^{2}+c \right )^{2}\, dx\]

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)^2,x)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{2} \left (\int x^{m} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- 2 a^{2} x^{2} x^{m} e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx + \int a^{4} x^{4} x^{m} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx\right ) \end {gather*}

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)**2,x)

[Out]

c**2*(Integral(x**m*exp(n*atanh(a*x)), x) + Integral(-2*a**2*x**2*x**m*exp(n*atanh(a*x)), x) + Integral(a**4*x
**4*x**m*exp(n*atanh(a*x)), x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*exp(n*atanh(a*x))*(c - a^2*c*x^2)^2,x)

[Out]

int(x^m*exp(n*atanh(a*x))*(c - a^2*c*x^2)^2, x)

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