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

Optimal. Leaf size=157 \[ -\frac{3 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}+\frac{2 \sqrt{2} \sqrt{1-a x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{1-a x} \sqrt{a x+1}}{a \sqrt{c-\frac{c}{a x}}} \]

[Out]

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

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Rubi [A]  time = 0.157716, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6134, 6129, 101, 157, 54, 215, 93, 206} \[ -\frac{3 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}+\frac{2 \sqrt{2} \sqrt{1-a x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{1-a x} \sqrt{a x+1}}{a \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 - a*x]*Sqrt[1 + a*x])/(a*Sqrt[c - c/(a*x)])) - (3*Sqrt[1 - a*x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*S
qrt[c - c/(a*x)]*Sqrt[x]) + (2*Sqrt[2]*Sqrt[1 - a*x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(3/2
)*Sqrt[c - c/(a*x)]*Sqrt[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 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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^{\tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\frac{\sqrt{1-a x} \int \frac{e^{\tanh ^{-1}(a x)} \sqrt{x}}{\sqrt{1-a x}} \, dx}{\sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{\sqrt{1-a x} \int \frac{\sqrt{x} \sqrt{1+a x}}{1-a x} \, dx}{\sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=-\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{1-a x} \int \frac{\frac{1}{2}+\frac{3 a x}{2}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=-\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}-\frac{\left (3 \sqrt{1-a x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a \sqrt{c-\frac{c}{a x}} \sqrt{x}}+\frac{\left (2 \sqrt{1-a x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=-\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}-\frac{\left (3 \sqrt{1-a x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}+\frac{\left (4 \sqrt{1-a x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=-\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}-\frac{3 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{c-\frac{c}{a x}} \sqrt{x}}+\frac{2 \sqrt{2} \sqrt{1-a x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{a^{3/2} \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0644654, size = 105, normalized size = 0.67 \[ -\frac{\sqrt{1-a x} \left (\sqrt{a} \sqrt{x} \sqrt{a x+1}+3 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.157, size = 168, normalized size = 1.1 \begin{align*} -{\frac{x\sqrt{2}}{4\,c \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-3\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+4\,\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 ) \sqrt{a} \right ){a}^{-{\frac{3}{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((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x)

[Out]

-1/4*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(2*(-(a*x+1)*x)^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)-3*arctan(1/
2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)+4*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))*2^(1/2)/a^(3/2)/c/(a*x-1)/(-(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{a x + 1}{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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