3.542 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{(c-\frac{c}{a x})^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ \frac{3 \sqrt{a x+1} (1-a x)^{5/2}}{2 a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{3 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{9 (1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{2 \sqrt{2} a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\sqrt{a x+1} (1-a x)^{3/2}}{2 a^2 x \left (c-\frac{c}{a x}\right )^{5/2}} \]

[Out]

((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(2*a^2*(c - c/(a*x))^(5/2)*x) + (3*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(2*a^3*(c -
c/(a*x))^(5/2)*x^2) + (3*(1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2)) - (9*
(1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(2*Sqrt[2]*a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/
2))

________________________________________________________________________________________

Rubi [A]  time = 0.189436, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6134, 6129, 98, 154, 157, 54, 215, 93, 206} \[ \frac{3 \sqrt{a x+1} (1-a x)^{5/2}}{2 a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{3 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{9 (1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{2 \sqrt{2} a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\sqrt{a x+1} (1-a x)^{3/2}}{2 a^2 x \left (c-\frac{c}{a x}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(5/2)),x]

[Out]

((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(2*a^2*(c - c/(a*x))^(5/2)*x) + (3*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(2*a^3*(c -
c/(a*x))^(5/2)*x^2) + (3*(1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2)) - (9*
(1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(2*Sqrt[2]*a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/
2))

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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

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)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx &=\frac{(1-a x)^{5/2} \int \frac{e^{-\tanh ^{-1}(a x)} x^{5/2}}{(1-a x)^{5/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2} \int \frac{x^{5/2}}{(1-a x)^2 \sqrt{1+a x}} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}-\frac{(1-a x)^{5/2} \int \frac{\sqrt{x} \left (\frac{3}{2}+3 a x\right )}{(1-a x) \sqrt{1+a x}} \, dx}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{3 (1-a x)^{5/2} \sqrt{1+a x}}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1-a x)^{5/2} \int \frac{-\frac{3 a}{2}-3 a^2 x}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{3 (1-a x)^{5/2} \sqrt{1+a x}}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{\left (3 (1-a x)^{5/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}-\frac{\left (9 (1-a x)^{5/2}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{4 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{3 (1-a x)^{5/2} \sqrt{1+a x}}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{\left (3 (1-a x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}-\frac{\left (9 (1-a x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{3 (1-a x)^{5/2} \sqrt{1+a x}}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{3 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}-\frac{9 (1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{2 \sqrt{2} a^{7/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.167171, size = 127, normalized size = 0.61 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{a x+1} (3-2 a x)-12 (a x-1) \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+9 \sqrt{2} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{4 a^{3/2} c^2 \sqrt{x} \sqrt{1-a x} \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(5/2)),x]

[Out]

(2*Sqrt[a]*Sqrt[x]*(3 - 2*a*x)*Sqrt[1 + a*x] - 12*(-1 + a*x)*ArcSinh[Sqrt[a]*Sqrt[x]] + 9*Sqrt[2]*(-1 + a*x)*A
rcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(4*a^(3/2)*c^2*Sqrt[c - c/(a*x)]*Sqrt[x]*Sqrt[1 - a*x])

________________________________________________________________________________________

Maple [A]  time = 0.152, size = 276, normalized size = 1.3 \begin{align*}{\frac{x\sqrt{2}}{8\,{c}^{3} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x-6\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-6\,{a}^{2}\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}x+9\,{a}^{3/2}\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+6\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}-9\,\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(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x)

[Out]

1/8*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(4*(-(a*x+1)*x)^(1/2)*a^(5/2)*2^(1/2)*(-1/a)^(1/2)*x-6*(-(a*x+1
)*x)^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)-6*a^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+9*a^(3/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a-3*a*x-1)/(a*x-1))*x+6*arctan(1/2/a^(1/2)*(2*
a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)-9*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^3/(a*x-1)^2/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(5/2)), x)

________________________________________________________________________________________

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(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{\frac{5}{2}} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(5/2),x)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(5/2)), x)