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

Optimal. Leaf size=127 \[ -\frac{x (1-a x)}{\sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}+\frac{5 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{1-a x}}{a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.203986, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6134, 6128, 881, 848, 47, 54, 215} \[ -\frac{x (1-a x)}{\sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}+\frac{5 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{1-a x}}{a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]),x]

[Out]

(-5*Sqrt[1 - a*x])/(a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x]) - (x*(1 - a*x))/(Sqrt[c - c/(a*x)]*Sqrt[1 - a^2*x^2]) +
(5*Sqrt[1 - a*x]*ArcSinh[Sqrt[a]*Sqrt[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 6128

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

Rule 881

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e^2*(d +
e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(e*f*(p + 1) - d*g*(2*n + p
+ 3))/(g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m,
n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1]
&& IntegerQ[2*p]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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]

Rubi steps

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

Mathematica [A]  time = 0.0602446, size = 86, normalized size = 0.68 \[ \frac{\sqrt{1-a x} \left (5 \sqrt{a x+1} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-\sqrt{a} \sqrt{x} (a x+5)\right )}{a^{3/2} \sqrt{x} \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.142, size = 143, normalized size = 1.1 \begin{align*} -{\frac{x}{2\, \left ( ax-1 \right ) \left ( ax+1 \right ) c}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+5\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) xa+10\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}+5\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(c*(a*x-1)/a/x)^(1/2)*x/a^(1/2)/c*(2*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+5*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1
)*x)^(1/2))*x*a+10*a^(1/2)*(-(a*x+1)*x)^(1/2)+5*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2)))*(-a^2*x^2+1)
^(1/2)/(a*x+1)/(-(a*x+1)*x)^(1/2)/(a*x-1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.30044, size = 598, normalized size = 4.71 \begin{align*} \left [-\frac{5 \,{\left (a^{2} x^{2} - 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 \,{\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{3} c x^{2} - a c\right )}}, \frac{5 \,{\left (a^{2} x^{2} - 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 \,{\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{3} c x^{2} - a c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(5*(a^2*x^2 - 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*(a^2*x^2 + 5*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3
*c*x^2 - a*c), 1/2*(5*(a^2*x^2 - 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*(a^2*x^2 + 5*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c*x^2 - a*c)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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