3.1357 \(\int e^{n \tanh ^{-1}(a x)} x^m (c-a^2 c x^2) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6150, 133} \[ \frac {c x^{m+1} F_1\left (m+1;\frac {n-2}{2},-\frac {n}{2}-1;m+2;a x,-a x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 1.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

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),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (a^{2} c x^{2} - c\right )} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

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),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} x^{m} \left (-a^{2} c \,x^{2}+c \right )\, dx \]

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),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (a^{2} c x^{2} - c\right )} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

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),x, algorithm="maxima")

[Out]

-integrate((a^2*c*x^2 - c)*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 \[ \int x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-a^2\,c\,x^2\right ) \,d x \]

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),x)

[Out]

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

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

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),x)

[Out]

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

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