3.248 \(\int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx\)

Optimal. Leaf size=177 \[ \frac{2 a x \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{6 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{3 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}} \]

[Out]

(-6*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/((a - x^(-1))*Sqrt[c - a*c*x]) + (2*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))
^(3/2)*x)/((a - x^(-1))*Sqrt[c - a*c*x]) - (3*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a
]*Sqrt[1 + 1/(a*x)])])/(Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x])

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Rubi [A]  time = 0.180234, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{2 a x \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{6 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{3 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/((a - x^(-1))*Sqrt[c - a*c*x]) + (2*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))
^(3/2)*x)/((a - x^(-1))*Sqrt[c - a*c*x]) - (3*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a
]*Sqrt[1 + 1/(a*x)])])/(Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*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 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^{3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a x}} \sqrt{x}\right ) \int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}} \sqrt{x}} \, dx}{\sqrt{c-a c x}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{3/2} \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 a \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{\left (6 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=-\frac{6 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}+\frac{2 a \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{\left (3 \sqrt{1-\frac{1}{a x}}\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 )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=-\frac{6 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}+\frac{2 a \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{\left (6 \sqrt{1-\frac{1}{a x}}\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 )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=-\frac{6 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}+\frac{2 a \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-a c x}}-\frac{3 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.143338, size = 116, normalized size = 0.66 \[ \frac{x \sqrt{1-\frac{1}{a x}} \left (2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (a x-2)-3 \sqrt{2} \sqrt{\frac{1}{x}} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{\sqrt{a} (a x-1) \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.178, size = 135, normalized size = 0.8 \begin{align*}{\frac{1}{a \left ( ax+1 \right ) }\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac-2\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}-3\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c+4\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63113, size = 682, normalized size = 3.85 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \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 ) - 4 \,{\left (a^{2} x^{2} - a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac{2 \,{\left (a^{2} x^{2} - a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}} - \frac{3 \, \sqrt{2}{\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \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 )}{\sqrt{c}}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.20042, size = 115, normalized size = 0.65 \begin{align*} \frac{3 \, \sqrt{2} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - 2 \, \sqrt{-a c x - c} + \frac{2 \, \sqrt{-a c x - c} c}{a c x - c}}{a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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