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

Optimal. Leaf size=61 \[ -\frac {2^{-p} (1-a x)^p (c-a c x)^{p+1} \, _2F_1\left (p,2 p+1;2 (p+1);\frac {1}{2} (1-a x)\right )}{a c (2 p+1)} \]

[Out]

-(-a*x+1)^p*(-a*c*x+c)^(1+p)*hypergeom([p, 1+2*p],[2+2*p],-1/2*a*x+1/2)/(2^p)/a/c/(1+2*p)

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Rubi [A]  time = 0.06, antiderivative size = 61, 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, 69} \[ -\frac {2^{-p} (1-a x)^p (c-a c x)^{p+1} \, _2F_1\left (p,2 p+1;2 (p+1);\frac {1}{2} (1-a x)\right )}{a c (2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - a*x)^p*(c - a*c*x)^(1 + p)*Hypergeometric2F1[p, 1 + 2*p, 2*(1 + p), (1 - a*x)/2])/(2^p*a*c*(1 + 2*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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

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} (c-a c x)^{2 p} \, dx\\ &=-\frac {2^{-p} (1-a x)^p (c-a c x)^{1+p} \, _2F_1\left (p,1+2 p;2 (1+p);\frac {1}{2} (1-a x)\right )}{a c (1+2 p)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 56, normalized size = 0.92 \[ -\frac {2^{-p} (1-a x)^{p+1} (c-a c x)^p \, _2F_1\left (p,2 p+1;2 p+2;\frac {1}{2}-\frac {a x}{2}\right )}{2 a p+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - a*x)^(1 + p)*(c - a*c*x)^p*Hypergeometric2F1[p, 1 + 2*p, 2 + 2*p, 1/2 - (a*x)/2])/(2^p*(a + 2*a*p)))

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (-a c x +c \right )^{p} {\mathrm e}^{-2 p \arctanh \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{-2\,p\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \]

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]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (a x - 1\right )\right )^{p} e^{- 2 p \operatorname {atanh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(a*x - 1))**p*exp(-2*p*atanh(a*x)), x)

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