3.14.8 \(\int e^{n \tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [1308]

Optimal. Leaf size=68 \[ -\frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \]

[Out]

-2^(2+1/2*n)*c*(-a*x+1)^(2-1/2*n)*hypergeom([-1-1/2*n, 2-1/2*n],[3-1/2*n],-1/2*a*x+1/2)/a/(4-n)

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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6275, 71} \begin {gather*} -\frac {c 2^{\frac {n}{2}+2} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-1,2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(2 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[-1 - n/2, 2 - n/2, 3 - n/2, (1 - a*x)/2])/(a*(4 - n)))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, 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)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x)^{1-\frac {n}{2}} (1+a x)^{1+\frac {n}{2}} \, dx\\ &=-\frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 65, normalized size = 0.96 \begin {gather*} \frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (-4+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(2 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[-1 - n/2, 2 - n/2, 3 - n/2, (1 - a*x)/2])/(a*(-4 + n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-integrate((a^2*c*x^2 - c)*(-(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))*(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(-(a^2*c*x^2 - c)*(-(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 \left (\int a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(Integral(a**2*x**2*exp(n*atanh(a*x)), x) + Integral(-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))*(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-a^2\,c\,x^2\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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