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

Optimal. Leaf size=143 \[ \frac {\sqrt {\frac {1}{a x}+1} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} (c-a c x)^p \, _2F_1\left (\frac {1}{2}-p,-p;1-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (p+1) \sqrt {1-\frac {1}{a x}}}+\frac {x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} (c-a c x)^p}{p+1} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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]

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*}

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Mathematica [A]  time = 0.10, size = 131, normalized size = 0.92 \[ \frac {\sqrt {\frac {1}{a x}+1} \left (\frac {a x-1}{a x+1}\right )^{-p} (c-a c x)^p \left (\sqrt {\frac {a x-1}{a x+1}} \, _2F_1\left (\frac {1}{2}-p,-p;1-p;\frac {2}{a x+1}\right )+p (a x-1) \left (\frac {a x-1}{a x+1}\right )^p\right )}{a p (p+1) \sqrt {1-\frac {1}{a x}}} \]

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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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )} {\left (-a c x + c\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, x\right ) \]

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

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)

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maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a c x +c \right )^{p}}{\sqrt {\frac {a x -1}{a x +1}}}\, dx \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a\,c\,x\right )}^p}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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