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

Optimal. Leaf size=37 \[ \frac{(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0346184, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6130, 23, 32} \[ \frac{(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6130

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 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)} (c-a c x)^p \, dx &=\int (1-a x)^{-p} (1+a x)^p (c-a c x)^p \, dx\\ &=\left ((1-a x)^{-p} (c-a c x)^p\right ) \int (1+a x)^p \, dx\\ &=\frac{(1-a x)^{-p} (1+a x)^{1+p} (c-a c x)^p}{a (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0236089, size = 32, normalized size = 0.86 \[ \frac{(a x+1) (c-a c x)^p e^{2 p \tanh ^{-1}(a x)}}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax+1 \right ){{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{p}}{a \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988049, size = 41, normalized size = 1.11 \begin{align*} \frac{{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )}{\left (a x + 1\right )}^{p}}{a{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.90134, size = 81, normalized size = 2.19 \begin{align*} \frac{{\left (a x + 1\right )}{\left (-a c x + c\right )}^{p} \left (\frac{a x + 1}{a x - 1}\right )^{p}}{a p + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError

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

[Out]

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