3.258 \(\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\sqrt{2} \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

[Out]

-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/((x^(-1))^
(3/2)*(c - a*c*x)^(3/2)))

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Rubi [A]  time = 0.180051, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6176, 6181, 93, 206} \[ -\frac{\sqrt{2} \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]

[Out]

-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/((x^(-1))^
(3/2)*(c - a*c*x)^(3/2)))

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 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{\left (2 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{\sqrt{2} \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0475276, size = 76, normalized size = 1. \[ -\frac{\sqrt{2} \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]

[Out]

-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/((x^(-1))^
(3/2)*(c - a*c*x)^(3/2)))

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Maple [A]  time = 0.151, size = 78, normalized size = 1. \begin{align*} -{\frac{ \left ( ax+1 \right ) \sqrt{2}}{ \left ( ax-1 \right ) a}\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{-c \left ( ax-1 \right ) }\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}{c}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x)

[Out]

-((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))/(a*
x-1)/(-c*(a*x+1))^(1/2)/c^(3/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (-a c x + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(3/2), x)

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Fricas [A]  time = 1.92783, size = 352, normalized size = 4.63 \begin{align*} \left [\frac{\sqrt{2} \sqrt{-\frac{1}{c}} \log \left (-\frac{a^{2} x^{2} + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{-\frac{1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, a c}, -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt{c}}\right )}{a c^{\frac{3}{2}}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x, algorithm="fricas")

[Out]

[1/2*sqrt(2)*sqrt(-1/c)*log(-(a^2*x^2 + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt(-1
/c) + 2*a*x - 3)/(a^2*x^2 - 2*a*x + 1))/(a*c), -sqrt(2)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x +
1))/((a*x - 1)*sqrt(c)))/(a*c^(3/2))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.17697, size = 86, normalized size = 1.13 \begin{align*} \frac{{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{a \sqrt{c}} - \frac{\sqrt{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{c}}\right )}{a \sqrt{c}}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x, algorithm="giac")

[Out]

(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(a*sqrt(c)) - sqrt(2)*arctan(sqrt(-c)/sqrt(c))/(a*sqrt(c
)))*abs(c)*sgn(a*x + 1)/c^2