3.543 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{(c-\frac{c}{a x})^{7/2}} \, dx\)

Optimal. Leaf size=252 \[ -\frac{35 \sqrt{a x+1} (1-a x)^{7/2}}{16 a^4 x^3 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{15 \sqrt{a x+1} (1-a x)^{5/2}}{16 a^3 x^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{5 (1-a x)^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{115 (1-a x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{16 \sqrt{2} a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\sqrt{a x+1} (1-a x)^{3/2}}{4 a^2 x \left (c-\frac{c}{a x}\right )^{7/2}} \]

[Out]

((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(4*a^2*(c - c/(a*x))^(7/2)*x) - (15*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(16*a^3*(c
- c/(a*x))^(7/2)*x^2) - (35*(1 - a*x)^(7/2)*Sqrt[1 + a*x])/(16*a^4*(c - c/(a*x))^(7/2)*x^3) - (5*(1 - a*x)^(7/
2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2)) + (115*(1 - a*x)^(7/2)*ArcTanh[(Sqrt[2]*Sqr
t[a]*Sqrt[x])/Sqrt[1 + a*x]])/(16*Sqrt[2]*a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2))

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Rubi [A]  time = 0.204563, antiderivative size = 252, 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, 98, 149, 154, 157, 54, 215, 93, 206} \[ -\frac{35 \sqrt{a x+1} (1-a x)^{7/2}}{16 a^4 x^3 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{15 \sqrt{a x+1} (1-a x)^{5/2}}{16 a^3 x^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{5 (1-a x)^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{115 (1-a x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{16 \sqrt{2} a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\sqrt{a x+1} (1-a x)^{3/2}}{4 a^2 x \left (c-\frac{c}{a x}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(7/2)),x]

[Out]

((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(4*a^2*(c - c/(a*x))^(7/2)*x) - (15*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(16*a^3*(c
- c/(a*x))^(7/2)*x^2) - (35*(1 - a*x)^(7/2)*Sqrt[1 + a*x])/(16*a^4*(c - c/(a*x))^(7/2)*x^3) - (5*(1 - a*x)^(7/
2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2)) + (115*(1 - a*x)^(7/2)*ArcTanh[(Sqrt[2]*Sqr
t[a]*Sqrt[x])/Sqrt[1 + a*x]])/(16*Sqrt[2]*a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2))

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 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^{-\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{7/2}} \, dx &=\frac{(1-a x)^{7/2} \int \frac{e^{-\tanh ^{-1}(a x)} x^{7/2}}{(1-a x)^{7/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{7/2} \int \frac{x^{7/2}}{(1-a x)^3 \sqrt{1+a x}} \, dx}{\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x}-\frac{(1-a x)^{7/2} \int \frac{x^{3/2} \left (\frac{5}{2}+5 a x\right )}{(1-a x)^2 \sqrt{1+a x}} \, dx}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x}-\frac{15 (1-a x)^{5/2} \sqrt{1+a x}}{16 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2}-\frac{(1-a x)^{7/2} \int \frac{\sqrt{x} \left (-\frac{45 a}{4}-\frac{35 a^2 x}{2}\right )}{(1-a x) \sqrt{1+a x}} \, dx}{8 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x}-\frac{15 (1-a x)^{5/2} \sqrt{1+a x}}{16 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2}-\frac{35 (1-a x)^{7/2} \sqrt{1+a x}}{16 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}+\frac{(1-a x)^{7/2} \int \frac{\frac{35 a^2}{4}+20 a^3 x}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{8 a^6 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x}-\frac{15 (1-a x)^{5/2} \sqrt{1+a x}}{16 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2}-\frac{35 (1-a x)^{7/2} \sqrt{1+a x}}{16 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}-\frac{\left (5 (1-a x)^{7/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}+\frac{\left (115 (1-a x)^{7/2}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{32 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x}-\frac{15 (1-a x)^{5/2} \sqrt{1+a x}}{16 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2}-\frac{35 (1-a x)^{7/2} \sqrt{1+a x}}{16 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}-\frac{\left (5 (1-a x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}+\frac{\left (115 (1-a x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{16 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{3/2} \sqrt{1+a x}}{4 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x}-\frac{15 (1-a x)^{5/2} \sqrt{1+a x}}{16 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2}-\frac{35 (1-a x)^{7/2} \sqrt{1+a x}}{16 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}-\frac{5 (1-a x)^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{9/2} \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}+\frac{115 (1-a x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{16 \sqrt{2} a^{9/2} \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.216278, size = 139, normalized size = 0.55 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{a x+1} \left (16 a^2 x^2-55 a x+35\right )+160 (a x-1)^2 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-115 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{32 a^{3/2} c^3 \sqrt{x} (1-a x)^{3/2} \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(7/2)),x]

[Out]

(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(35 - 55*a*x + 16*a^2*x^2) + 160*(-1 + a*x)^2*ArcSinh[Sqrt[a]*Sqrt[x]] - 115*
Sqrt[2]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(32*a^(3/2)*c^3*Sqrt[c - c/(a*x)]*Sqrt[
x]*(1 - a*x)^(3/2))

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Maple [A]  time = 0.151, size = 390, normalized size = 1.6 \begin{align*}{\frac{x\sqrt{2}}{64\,{c}^{4} \left ( ax-1 \right ) ^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 32\,\sqrt{- \left ( ax+1 \right ) x}{a}^{7/2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}-110\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x-80\,{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}+115\,{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}+70\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}+160\,{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-230\,{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-80\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+115\,\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)^(7/2),x)

[Out]

1/64*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(32*(-(a*x+1)*x)^(1/2)*a^(7/2)*2^(1/2)*(-1/a)^(1/2)*x^2-110*(-
(a*x+1)*x)^(1/2)*a^(5/2)*2^(1/2)*(-1/a)^(1/2)*x-80*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+115*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+70*(-(a*x
+1)*x)^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)+160*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-230*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-80*arctan(1/2/a^(1
/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)+115*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^4/(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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{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)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(7/2)), 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(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(7/2),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(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(7/2),x)

[Out]

Timed out

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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{7}{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)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(7/2)), x)