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

Optimal. Leaf size=249 \[ -\frac{9 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{51 (1-a x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{4 \sqrt{2} a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{21 \sqrt{a x+1} (1-a x)^{3/2}}{8 a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9 (a x+1)^{3/2} \sqrt{1-a x}}{8 a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{(a x+1)^{3/2}}{2 a \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{3/2}} \]

[Out]

(-21*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(8*a^2*(c - c/(a*x))^(3/2)*x) + (1 + a*x)^(3/2)/(2*a*(c - c/(a*x))^(3/2)*S
qrt[1 - a*x]) - (9*Sqrt[1 - a*x]*(1 + a*x)^(3/2))/(8*a^2*(c - c/(a*x))^(3/2)*x) - (9*(1 - a*x)^(3/2)*ArcSinh[S
qrt[a]*Sqrt[x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2)) + (51*(1 - a*x)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])
/Sqrt[1 + a*x]])/(4*Sqrt[2]*a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2))

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Rubi [A]  time = 0.205266, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6134, 6129, 97, 149, 154, 157, 54, 215, 93, 206} \[ -\frac{9 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{51 (1-a x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{4 \sqrt{2} a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{21 \sqrt{a x+1} (1-a x)^{3/2}}{8 a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9 (a x+1)^{3/2} \sqrt{1-a x}}{8 a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{(a x+1)^{3/2}}{2 a \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(8*a^2*(c - c/(a*x))^(3/2)*x) + (1 + a*x)^(3/2)/(2*a*(c - c/(a*x))^(3/2)*S
qrt[1 - a*x]) - (9*Sqrt[1 - a*x]*(1 + a*x)^(3/2))/(8*a^2*(c - c/(a*x))^(3/2)*x) - (9*(1 - a*x)^(3/2)*ArcSinh[S
qrt[a]*Sqrt[x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2)) + (51*(1 - a*x)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])
/Sqrt[1 + a*x]])/(4*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 149

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

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^{3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=\frac{(1-a x)^{3/2} \int \frac{e^{3 \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} (1+a x)^{3/2}}{(1-a x)^3} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{(1+a x)^{3/2}}{2 a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1-a x}}-\frac{(1-a x)^{3/2} \int \frac{\sqrt{x} \sqrt{1+a x} \left (\frac{3}{2}+3 a x\right )}{(1-a x)^2} \, dx}{2 a \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{(1+a x)^{3/2}}{2 a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1-a x}}-\frac{9 \sqrt{1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{(1-a x)^{3/2} \int \frac{\sqrt{1+a x} \left (-\frac{9 a}{4}-\frac{21 a^2 x}{2}\right )}{\sqrt{x} (1-a x)} \, dx}{4 a^3 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{21 (1-a x)^{3/2} \sqrt{1+a x}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{(1+a x)^{3/2}}{2 a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1-a x}}-\frac{9 \sqrt{1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{(1-a x)^{3/2} \int \frac{\frac{15 a^2}{2}+18 a^3 x}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{4 a^4 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{21 (1-a x)^{3/2} \sqrt{1+a x}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{(1+a x)^{3/2}}{2 a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1-a x}}-\frac{9 \sqrt{1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{\left (9 (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 (51 (1-a x)^{3/2}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{21 (1-a x)^{3/2} \sqrt{1+a x}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{(1+a x)^{3/2}}{2 a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1-a x}}-\frac{9 \sqrt{1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{\left (9 (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 (51 (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 )}{4 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{21 (1-a x)^{3/2} \sqrt{1+a x}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}+\frac{(1+a x)^{3/2}}{2 a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1-a x}}-\frac{9 \sqrt{1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{9 (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{51 (1-a x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{4 \sqrt{2} a^{5/2} \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.140049, size = 139, normalized size = 0.56 \[ -\frac{-2 \sqrt{a} \sqrt{x} \sqrt{a x+1} \left (4 a^2 x^2-23 a x+15\right )-72 (a x-1)^2 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+51 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{8 a^{3/2} c \sqrt{x} (1-a x)^{3/2} \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(15 - 23*a*x + 4*a^2*x^2) - 72*(-1 + a*x)^2*ArcSinh[Sqrt[a]*Sqrt[x]] + 51*S
qrt[2]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(8*a^(3/2)*c*Sqrt[c - c/(a*x)]*Sqrt[x]*(
1 - a*x)^(3/2))

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Maple [B]  time = 0.148, size = 390, normalized size = 1.6 \begin{align*}{\frac{x\sqrt{2}}{16\,{c}^{2} \left ( ax-1 \right ) ^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 8\,\sqrt{- \left ( ax+1 \right ) x}{a}^{7/2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}-36\,{a}^{3}\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}-46\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x+51\,{a}^{5/2}\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}^{2}+72\,{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+30\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-102\,{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-36\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+51\,\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)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(3/2),x)

[Out]

1/16*(c*(a*x-1)/a/x)^(1/2)*x/a^(3/2)/c^2*(-a^2*x^2+1)^(1/2)*(8*(-(a*x+1)*x)^(1/2)*a^(7/2)*2^(1/2)*(-1/a)^(1/2)
*x^2-36*a^3*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*2^(1/2)*(-1/a)^(1/2)*x^2-46*(-(a*x+1)*x)^(1/2)*a^
(5/2)*2^(1/2)*(-1/a)^(1/2)*x+51*a^(5/2)*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+
72*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+30*(-(a*x+1)*x)^(1/2)*a^(3/2)*2
^(1/2)*(-1/a)^(1/2)-102*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-36*arctan(
1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)+51*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)^3/(-(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{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\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)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(3/2),x)

[Out]

Timed out

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

[Out]

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