3.516 \(\int \frac{e^{\tanh ^{-1}(a x)}}{(c-\frac{c}{a x})^{3/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{5 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{7 (1-a x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{2} a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 \sqrt{a x+1} (1-a x)^{3/2}}{a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\sqrt{a x+1} \sqrt{1-a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.186758, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6134, 6129, 97, 154, 157, 54, 215, 93, 206} \[ \frac{5 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{7 (1-a x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{2} a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 \sqrt{a x+1} (1-a x)^{3/2}}{a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\sqrt{a x+1} \sqrt{1-a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - c/(a*x))^(3/2),x]

[Out]

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

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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 )^{3/2}} \, dx &=\frac{(1-a x)^{3/2} \int \frac{e^{\tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{(1-a x)^{3/2} \int \frac{x^{3/2} \sqrt{1+a x}}{(1-a x)^2} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{(1-a x)^{3/2} \int \frac{\sqrt{x} \left (\frac{3}{2}+2 a x\right )}{(1-a x) \sqrt{1+a x}} \, dx}{a \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{(1-a x)^{3/2} \int \frac{-a-\frac{5 a^2 x}{2}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a^3 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{\left (5 (1-a x)^{3/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}-\frac{\left (7 (1-a x)^{3/2}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{2 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{\left (5 (1-a x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}-\frac{\left (7 (1-a x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{\sqrt{1-a x} \sqrt{1+a x}}{a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{5 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}-\frac{7 (1-a x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{\sqrt{2} a^{5/2} \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.149247, size = 126, normalized size = 0.64 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{a x+1} (a x-2)+10 (a x-1) \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-7 \sqrt{2} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{2 a^{3/2} c \sqrt{x} \sqrt{1-a x} \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(c - c/(a*x))^(3/2),x]

[Out]

(2*Sqrt[a]*Sqrt[x]*(-2 + a*x)*Sqrt[1 + a*x] + 10*(-1 + a*x)*ArcSinh[Sqrt[a]*Sqrt[x]] - 7*Sqrt[2]*(-1 + a*x)*Ar
cTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(2*a^(3/2)*c*Sqrt[c - c/(a*x)]*Sqrt[x]*Sqrt[1 - a*x])

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Maple [A]  time = 0.152, size = 276, normalized size = 1.4 \begin{align*} -{\frac{x\sqrt{2}}{4\,{c}^{2} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x-5\,{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-4\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}+7\,{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+5\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}-7\,\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)^(3/2),x)

[Out]

-1/4*(c*(a*x-1)/a/x)^(1/2)*x/a^(3/2)/c^2*(-a^2*x^2+1)^(1/2)*(2*(-(a*x+1)*x)^(1/2)*a^(5/2)*2^(1/2)*(-1/a)^(1/2)
*x-5*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-4*(-(a*x+1)*x)^(1/2)*a^(3/2)*
2^(1/2)*(-1/a)^(1/2)+7*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+5*arctan(1/
2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)-7*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*x-1)^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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 3.13346, size = 1146, normalized size = 5.79 \begin{align*} \left [-\frac{7 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, 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 ) + 10 \,{\left (a^{2} x^{2} - 2 \, 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 ) + 8 \,{\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac{7 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, 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 ) - 10 \,{\left (a^{2} x^{2} - 2 \, 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 ) - 4 \,{\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \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)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(7*sqrt(2)*(a^2*x^2 - 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)*sqrt(-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)) + 10*(
a^2*x^2 - 2*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)*sq
rt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(a^2*x^2 - 2*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c
^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/4*(7*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(2*sqrt(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)) - 10*(a^2*x^2 - 2*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)) - 4*(a^2*x^2 - 2*a*x)*sqrt(
-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

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

[Out]

Integral((a*x + 1)/((-c*(-1 + 1/(a*x)))**(3/2)*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}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

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