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]

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

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 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]

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]
)

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.56, size = 37, normalized size = 1.00 \[ \frac {{\left (a x + 1\right )} {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}}{a p + a} \]

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

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)

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maple [A]  time = 0.03, size = 32, normalized size = 0.86 \[ \frac {\left (a x +1\right ) {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (-a c x +c \right )^{p}}{a \left (1+p \right )} \]

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 = 0.40, size = 30, normalized size = 0.81 \[ \frac {{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )} {\left (a x + 1\right )}^{p}}{a {\left (p + 1\right )}} \]

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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mupad [B]  time = 0.89, size = 37, normalized size = 1.00 \[ \frac {{\left (c-a\,c\,x\right )}^p\,{\left (a\,x+1\right )}^{p+1}}{a\,{\left (1-a\,x\right )}^p\,\left (p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {x}{c} & \text {for}\: a = 0 \wedge p = -1 \\c^{p} x & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{2 \operatorname {atanh}{\left (a x \right )}} - e^{2 \operatorname {atanh}{\left (a x \right )}}}\, dx}{c} & \text {for}\: p = -1 \\\frac {a x \left (- a c x + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{a p + a} + \frac {\left (- a c x + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{a p + a} & \text {otherwise} \end {cases} \]

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]

Piecewise((x/c, Eq(a, 0) & Eq(p, -1)), (c**p*x, Eq(a, 0)), (-Integral(1/(a*x*exp(2*atanh(a*x)) - exp(2*atanh(a
*x))), x)/c, Eq(p, -1)), (a*x*(-a*c*x + c)**p*exp(2*p*atanh(a*x))/(a*p + a) + (-a*c*x + c)**p*exp(2*p*atanh(a*
x))/(a*p + a), True))

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