3.1365 \(\int e^{2 p \tanh ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\)

Optimal. Leaf size=49 \[ \frac{(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]

[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 = 0.0728544, antiderivative size = 49, 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} \[ \frac{(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*p*ArcTanh[a*x])*(c - a^2*c*x^2)^p,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 = 0.0248821, size = 36, normalized size = 0.73 \[ \frac{(a x+1) \left (c-a^2 c x^2\right )^p e^{2 p \tanh ^{-1}(a x)}}{2 a p+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.024, size = 38, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax+1 \right ){{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{ \left ( 1+2\,p \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.982954, size = 46, normalized size = 0.94 \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(exp(2*p*arctanh(a*x))*(-a^2*c*x^2+c)^p,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 = 2.14824, size = 89, normalized size = 1.82 \begin{align*} \frac{{\left (a x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x + 1}{a x - 1}\right )^{p}}{2 \, a p + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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(exp(2*p*arctanh(a*x))*(-a^2*c*x^2+c)^p,x, algorithm="giac")

[Out]

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