3.1366 \(\int e^{-2 p \tanh ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\)

Optimal. Leaf size=51 \[ -\frac{(1-a x)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0705453, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6143, 6140, 32} \[ -\frac{(1-a x)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6143

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c^IntPart[p]*(c + d*x^2)^Frac
Part[p])/(1 - a^2*x^2)^FracPart[p], Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x
] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 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])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0277983, size = 36, normalized size = 0.71 \[ \frac{(a x-1) \left (c-a^2 c x^2\right )^p e^{-2 p \tanh ^{-1}(a x)}}{2 a p+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 40, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax-1 \right ) \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{a \left ( 1+2\,p \right ){{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.9907, size = 49, normalized size = 0.96 \begin{align*} \frac{{\left (a \left (-c\right )^{p} x - \left (-c\right )^{p}\right )}{\left (a x - 1\right )}^{2 \, p}}{a{\left (2 \, p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.13192, size = 92, normalized size = 1.8 \begin{align*} \frac{{\left (a x - 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (2 \, a p + a\right )} \left (\frac{a x + 1}{a x - 1}\right )^{p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} c x^{2} + c\right )}^{p}}{\left (\frac{a x + 1}{a x - 1}\right )^{p}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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