### 3.157 $$\int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx$$

Optimal. Leaf size=143 $\frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2}-p} (c-a c x)^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,1-p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{a p (p+1) \sqrt{1-\frac{1}{a x}}}+\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{p+1}$

[Out]

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

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Rubi [A]  time = 0.157184, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6176, 6181, 94, 132} $\frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2}-p} (c-a c x)^p \, _2F_1\left (\frac{1}{2}-p,-p;1-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{a p (p+1) \sqrt{1-\frac{1}{a x}}}+\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{p+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]*(c - a*c*x)^p,x]

[Out]

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

Rule 6176

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

Rule 6181

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

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c -
a*d)*(e + f*x)))])/(((b*e - a*f)*(m + 1))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n), x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx &=\left (\left (1-\frac{1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^p x^p \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{x}\right )^p (c-a c x)^p\right ) \operatorname{Subst}\left (\int x^{-2-p} \left (1-\frac{x}{a}\right )^{-\frac{1}{2}+p} \sqrt{1+\frac{x}{a}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x (c-a c x)^p}{1+p}-\frac{\left (\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{x}\right )^p (c-a c x)^p\right ) \operatorname{Subst}\left (\int \frac{x^{-1-p} \left (1-\frac{x}{a}\right )^{-\frac{1}{2}+p}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a (1+p)}\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x (c-a c x)^p}{1+p}+\frac{\left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2}-p} \sqrt{1+\frac{1}{a x}} (c-a c x)^p \, _2F_1\left (\frac{1}{2}-p,-p;1-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{a p (1+p) \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0849435, size = 131, normalized size = 0.92 $\frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a x-1}{a x+1}\right )^{-p} (c-a c x)^p \left (\sqrt{\frac{a x-1}{a x+1}} \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,1-p,\frac{2}{a x+1}\right )+p (a x-1) \left (\frac{a x-1}{a x+1}\right )^p\right )}{a p (p+1) \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[1 + 1/(a*x)]*(c - a*c*x)^p*(p*(-1 + a*x)*((-1 + a*x)/(1 + a*x))^p + Sqrt[(-1 + a*x)/(1 + a*x)]*Hypergeom
etric2F1[1/2 - p, -p, 1 - p, 2/(1 + a*x)]))/(a*p*(1 + p)*Sqrt[1 - 1/(a*x)]*((-1 + a*x)/(1 + a*x))^p)

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Maple [F]  time = 0.392, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -acx+c \right ) ^{p}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a x + 1\right )}{\left (-a c x + c\right )}^{p} \sqrt{\frac{a x - 1}{a x + 1}}}{a x - 1}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^p,x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(a*x - 1))**p/sqrt((a*x - 1)/(a*x + 1)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^p,x, algorithm="giac")

[Out]

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