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

Optimal. Leaf size=154 \[ -\frac{2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}-\frac{2 \sqrt{a} \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{a} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.251498, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6134, 6129, 98, 157, 54, 215, 93, 206} \[ -\frac{2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}-\frac{2 \sqrt{a} \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{a} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6134

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

Rule 6129

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

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 157

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

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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

Mathematica [A]  time = 0.0468401, size = 103, normalized size = 0.67 \[ -\frac{2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{a x+1}+\sqrt{a} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-2 \sqrt{2} \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{\sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x] + Sqrt[a]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]] - 2*Sqrt[2]*Sqrt[a]*Sqrt[x]*Ar
cTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/Sqrt[1 - a*x]

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Maple [A]  time = 0.15, size = 166, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{2\,ax-2}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( -\arctan \left ({\frac{2\,ax+1}{2}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}xa+2\,\sqrt{- \left ( ax+1 \right ) x}\sqrt{a}\sqrt{2}\sqrt{-{a}^{-1}}+4\,\sqrt{a}\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ) x \right ){\frac{1}{\sqrt{a}}}{\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((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.95535, size = 949, normalized size = 6.16 \begin{align*} \left [\frac{2 \, \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^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) +{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a x - 1\right )}}, -\frac{2 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) -{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*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^2*x^2 + a*x)*s
qrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + (a*x - 1)*sqrt(-c
)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(
a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1), -(2*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sq
rt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (a*x - 1)*sqrt(c)*
arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 +
1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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