3.551 \(\int e^{-2 p \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\)

Optimal. Leaf size=93 \[ -\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{a x}+1\right )^{1-p} F_1\left (1-p;-2 p,2;2-p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a (1-p)} \]

[Out]

-4^p*(1+1/a/x)^(1-p)*(c-c/a/x)^p*AppellF1(1-p,-2*p,2,2-p,1/2*(a+1/x)/a,1+1/a/x)/a/(1-p)/((1-1/a/x)^p)

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Rubi [A]  time = 0.10, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6182, 6179, 136} \[ -\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{a x}+1\right )^{1-p} F_1\left (1-p;-2 p,2;2-p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a (1-p)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4^p*(1 + 1/(a*x))^(1 - p)*(c - c/(a*x))^p*AppellF1[1 - p, -2*p, 2, 2 - p, (a + x^(-1))/(2*a), 1 + 1/(a*x)])
/(a*(1 - p)*(1 - 1/(a*x))^p))

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 6179

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

Rule 6182

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

Rubi steps

\begin {align*} \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \int e^{-2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{2 p} \left (1+\frac {x}{a}\right )^{-p}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1-p} \left (c-\frac {c}{a x}\right )^p F_1\left (1-p;-2 p,2;2-p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (1-p)}\\ \end {align*}

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Mathematica [F]  time = 0.66, size = 0, normalized size = 0.00 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a c x - c}{a x}\right )^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{p}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (c -\frac {c}{a x}\right )^{p} {\mathrm e}^{-2 p \,\mathrm {arccoth}\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{-2\,p\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a\,x}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(-1 + 1/(a*x)))**p*exp(-2*p*acoth(a*x)), x)

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