3.14.63 \(\int e^{2 (1+p) \tanh ^{-1}(a x)} (1-a^2 x^2)^{-p} \, dx\) [1363]

Optimal. Leaf size=41 \[ \frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p} \]

[Out]

(-a*x+1)^(1-2*p)/a/(1-2*p)+1/a/p/((-a*x+1)^(2*p))

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Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6275, 45} \begin {gather*} \frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*(1 + p)*ArcTanh[a*x])/(1 - a^2*x^2)^p,x]

[Out]

(1 - a*x)^(1 - 2*p)/(a*(1 - 2*p)) + 1/(a*p*(1 - a*x)^(2*p))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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^{2 (1+p) \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{-p} \, dx &=\int (1-a x)^{-1-2 p} (1+a x) \, dx\\ &=\int \left (2 (1-a x)^{-1-2 p}-(1-a x)^{-2 p}\right ) \, dx\\ &=\frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.76 \begin {gather*} \frac {(1-a x)^{-2 p} (-1+p+a p x)}{a p (-1+2 p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*(1 + p)*ArcTanh[a*x])/(1 - a^2*x^2)^p,x]

[Out]

(-1 + p + a*p*x)/(a*p*(-1 + 2*p)*(1 - a*x)^(2*p))

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Maple [A]
time = 0.08, size = 59, normalized size = 1.44

method	result	size
gosper	\(-\frac {\left (a x -1\right ) \left (a p x +p -1\right ) {\mathrm e}^{2 \left (1+p \right ) \arctanh \left (a x \right )} \left (-a^{2} x^{2}+1\right )^{-p}}{a p \left (2 p -1\right ) \left (a x +1\right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*(1+p)*arctanh(a*x))/((-a^2*x^2+1)^p),x,method=_RETURNVERBOSE)

[Out]

-(a*x-1)*(a*p*x+p-1)*exp(2*(1+p)*arctanh(a*x))/a/p/(2*p-1)/(a*x+1)/((-a^2*x^2+1)^p)

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Maxima [A]
time = 0.28, size = 34, normalized size = 0.83 \begin {gather*} \frac {a p x + p - 1}{{\left (2 \, p^{2} - p\right )} {\left (-a x + 1\right )}^{2 \, p} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*(1+p)*arctanh(a*x))/((-a^2*x^2+1)^p),x, algorithm="maxima")

[Out]

(a*p*x + p - 1)/((2*p^2 - p)*(-a*x + 1)^(2*p)*a)

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Fricas [A]
time = 0.34, size = 81, normalized size = 1.98 \begin {gather*} -\frac {{\left (a^{2} p x^{2} - a x - p + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{p + 1}}{{\left (2 \, a p^{2} - a p + {\left (2 \, a^{2} p^{2} - a^{2} p\right )} x\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*(1+p)*arctanh(a*x))/((-a^2*x^2+1)^p),x, algorithm="fricas")

[Out]

-(a^2*p*x^2 - a*x - p + 1)*(-(a*x + 1)/(a*x - 1))^(p + 1)/((2*a*p^2 - a*p + (2*a^2*p^2 - a^2*p)*x)*(-a^2*x^2 +
1)^p)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{- p} e^{2 p \operatorname {atanh}{\left (a x \right )}} e^{2 \operatorname {atanh}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*(1+p)*atanh(a*x))/((-a**2*x**2+1)**p),x)

[Out]

Integral(exp(2*p*atanh(a*x))*exp(2*atanh(a*x))/(-(a*x - 1)*(a*x + 1))**p, 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(2*(1+p)*arctanh(a*x))/((-a^2*x^2+1)^p),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.12, size = 84, normalized size = 2.05 \begin {gather*} \frac {p\,{\left (a\,x+1\right )}^p-{\left (a\,x+1\right )}^p+a\,p\,x\,{\left (a\,x+1\right )}^p}{2\,a\,p^2\,{\left (1-a^2\,x^2\right )}^p\,{\left (1-a\,x\right )}^p-a\,p\,{\left (1-a^2\,x^2\right )}^p\,{\left (1-a\,x\right )}^p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*atanh(a*x)*(p + 1))/(1 - a^2*x^2)^p,x)

[Out]

(p*(a*x + 1)^p - (a*x + 1)^p + a*p*x*(a*x + 1)^p)/(2*a*p^2*(1 - a^2*x^2)^p*(1 - a*x)^p - a*p*(1 - a^2*x^2)^p*(
1 - a*x)^p)

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