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

Optimal. Leaf size=209 \[ \frac{23 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{4 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{2 \sqrt{1-\frac{1}{a x}}}+\frac{9 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-\frac{1}{a x}}} \]

[Out]

(9*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/(4*a*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^2)/
(2*Sqrt[1 - 1/(a*x)]) + (23*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(4*a^2*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[
2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a^2*Sqrt[1 - 1/(a*x)])

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Rubi [A]  time = 0.21892, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {6182, 6180, 98, 151, 156, 63, 208, 206} \[ \frac{23 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{4 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{2 \sqrt{1-\frac{1}{a x}}}+\frac{9 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/(4*a*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^2)/
(2*Sqrt[1 - 1/(a*x)]) + (23*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(4*a^2*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[
2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a^2*Sqrt[1 - 1/(a*x)])

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])

Rule 6180

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

Rule 98

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

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 e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^3 \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{2 \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{9}{2 a}-\frac{7 x}{2 a^2}}{x^2 \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{4 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{2 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\frac{23}{4 a^2}+\frac{9 x}{4 a^3}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{4 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{2 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}-\frac{\left (23 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{4 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{2 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}-\frac{\left (23 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{4 a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{4 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{2 \sqrt{1-\frac{1}{a x}}}+\frac{23 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{4 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.532059, size = 236, normalized size = 1.13 \[ \frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (2 a x+9) \sqrt{c-\frac{c}{a x}}}{a x-1}+23 \sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-16 \sqrt{2} \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )-23 \sqrt{c} \log (1-a x)+16 \sqrt{2} \sqrt{c} \log \left ((a x-1)^2\right )}{8 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(9 + 2*a*x))/(-1 + a*x) - 23*Sqrt[c]*Log[1 - a*x] + 16*Sqr
t[2]*Sqrt[c]*Log[(-1 + a*x)^2] + 23*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*
(-1 - a*x + 2*a^2*x^2)] - 16*Sqrt[2]*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]
*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)])/(8*a^2)

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Maple [A]  time = 0.176, size = 180, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax-1 \right ) x}{8\,ax+8}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 4\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x+18\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-16\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a}+23\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(4*a^(5/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*x
+18*((a*x+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)-16*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a
*x-1))*a^(1/2)+23*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2))/a^(5/2)/((a*x+1)*x)^(1/
2)/(1/a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} x}{\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)*x*(c-c/a/x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.20549, size = 1187, normalized size = 5.68 \begin{align*} \left [\frac{16 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 23 \,{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (2 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + 9 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{16 \,{\left (a^{3} x - a^{2}\right )}}, \frac{16 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{2}{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 23 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (2 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + 9 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{8 \,{\left (a^{3} x - a^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(16*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2
*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1))
+ 23*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a
*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(2*a^3*x^3 + 11*a^2*x^2 + 9*a*x)*sqrt((a*x - 1)/(a*x + 1)
)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2), 1/8*(16*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*
sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 23*(a*x - 1)*sqrt(-c
)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c
)) + 2*(2*a^3*x^3 + 11*a^2*x^2 + 9*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^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(1/((a*x-1)/(a*x+1))**(3/2)*x*(c-c/a/x)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} x}{\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)*x*(c-c/a/x)^(1/2),x, algorithm="giac")

[Out]

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