### 3.259 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx$$

Optimal. Leaf size=136 $\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{2 \sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}-\frac{a^2 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}{2 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}$

[Out]

-(a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/(2*(a - x^(-1))*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(
5/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(2*Sqrt[2]*(x^(-1))^(5/2)*(c - a*c*x)^(5/2))

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Rubi [A]  time = 0.192643, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6176, 6181, 94, 93, 206} $\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{2 \sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}-\frac{a^2 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}{2 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]

[Out]

-(a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/(2*(a - x^(-1))*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(
5/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(2*Sqrt[2]*(x^(-1))^(5/2)*(c - a*c*x)^(5/2))

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 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 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)^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}{2 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}+\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \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 )}{4 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}{2 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}+\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\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 )}{2 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}{2 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}+\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{2 \sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.114456, size = 116, normalized size = 0.85 $\frac{x \sqrt{1-\frac{1}{a x}} \left (\sqrt{2} \sqrt{\frac{1}{x}} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )-2 \sqrt{a} \sqrt{\frac{1}{a x}+1}\right )}{4 \sqrt{a} c^2 (a x-1) \sqrt{c-a c x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]

[Out]

(Sqrt[1 - 1/(a*x)]*x*(-2*Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(
-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(4*Sqrt[a]*c^2*(-1 + a*x)*Sqrt[c - a*c*x])

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Maple [A]  time = 0.175, size = 123, normalized size = 0.9 \begin{align*}{\frac{ax+1}{4\, \left ( ax-1 \right ) ^{2}a}\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{-c \left ( ax-1 \right ) } \left ( -\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) xac+\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) c+2\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \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)^(5/2),x)

[Out]

1/4*((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)
)*x*a*c+2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c+2*(-c*(a*x+1))^(1/2)*c^(1/2))/c^(7/2)/(a*x-1)
^2/(-c*(a*x+1))^(1/2)/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{5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(5/2), x)

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Fricas [A]  time = 1.89018, size = 664, normalized size = 4.88 \begin{align*} \left [-\frac{\sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac{\sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}\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)^(5/2),x, algorithm="fricas")

[Out]

[-1/8*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)
*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1
)/(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3), -1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(sqrt(2)*
sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/
(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]

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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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21439, size = 99, normalized size = 0.73 \begin{align*} \frac{{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{a c^{\frac{3}{2}}} - \frac{2 \, \sqrt{-a c x - c}}{{\left (a c x - c\right )} a c}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{4 \, 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)^(5/2),x, algorithm="giac")

[Out]

1/4*(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(a*c^(3/2)) - 2*sqrt(-a*c*x - c)/((a*c*x - c)*a*c))*
abs(c)*sgn(a*x + 1)/c^2