3.1309 \(\int e^{n \tanh ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=70 \[ -\frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-2,3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (6-n)} \]

[Out]

-2^(3+1/2*n)*c^2*(-a*x+1)^(3-1/2*n)*hypergeom([3-1/2*n, -2-1/2*n],[4-1/2*n],-1/2*a*x+1/2)/a/(6-n)

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Rubi [A]  time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 69} \[ -\frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-2,3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (6-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(3 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[-2 - n/2, 3 - n/2, 4 - n/2, (1 - a*x)/2])/(a*(6 - n))
)

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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 6140

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 )^2 \, dx &=c^2 \int (1-a x)^{2-\frac {n}{2}} (1+a x)^{2+\frac {n}{2}} \, dx\\ &=-\frac {2^{3+\frac {n}{2}} c^2 (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-2-\frac {n}{2},3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (6-n)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 67, normalized size = 0.96 \[ \frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-2,3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (n-6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(3 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[-2 - n/2, 3 - n/2, 4 - n/2, (1 - a*x)/2])/(a*(-6 + n))

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fricas [F]  time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-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)*((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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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