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

Optimal. Leaf size=172 \[ -\frac{107 x^2 \sqrt{c-\frac{c}{a x}}}{96 a^2}+\frac{149 x \sqrt{c-\frac{c}{a x}}}{64 a^3}-\frac{363 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{64 a^4}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a^4}-\frac{1}{4} x^4 \sqrt{c-\frac{c}{a x}}+\frac{17 x^3 \sqrt{c-\frac{c}{a x}}}{24 a} \]

[Out]

(149*Sqrt[c - c/(a*x)]*x)/(64*a^3) - (107*Sqrt[c - c/(a*x)]*x^2)/(96*a^2) + (17*Sqrt[c - c/(a*x)]*x^3)/(24*a)
- (Sqrt[c - c/(a*x)]*x^4)/4 - (363*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(64*a^4) + (4*Sqrt[2]*Sqrt[c]*A
rcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^4

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Rubi [A]  time = 0.352306, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 446, 98, 151, 156, 63, 208} \[ -\frac{107 x^2 \sqrt{c-\frac{c}{a x}}}{96 a^2}+\frac{149 x \sqrt{c-\frac{c}{a x}}}{64 a^3}-\frac{363 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{64 a^4}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a^4}-\frac{1}{4} x^4 \sqrt{c-\frac{c}{a x}}+\frac{17 x^3 \sqrt{c-\frac{c}{a x}}}{24 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(149*Sqrt[c - c/(a*x)]*x)/(64*a^3) - (107*Sqrt[c - c/(a*x)]*x^2)/(96*a^2) + (17*Sqrt[c - c/(a*x)]*x^3)/(24*a)
- (Sqrt[c - c/(a*x)]*x^4)/4 - (363*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(64*a^4) + (4*Sqrt[2]*Sqrt[c]*A
rcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^4

Rule 6133

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[(u*(c + d/x)^p*(1 + a*x)^(n/
2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &
&  !GtQ[c, 0]

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx &=\int \frac{\sqrt{c-\frac{c}{a x}} x^3 (1-a x)}{1+a x} \, dx\\ &=-\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2} x^4}{1+a x} \, dx}{c}\\ &=-\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2} x^3}{a+\frac{1}{x}} \, dx}{c}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{x^5 (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{\operatorname{Subst}\left (\int \frac{\frac{17 c^2}{2}-\frac{15 c^2 x}{2 a}}{x^4 (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=\frac{17 \sqrt{c-\frac{c}{a x}} x^3}{24 a}-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4+\frac{\operatorname{Subst}\left (\int \frac{\frac{107 c^3}{4}-\frac{85 c^3 x}{4 a}}{x^3 (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{12 a c^2}\\ &=-\frac{107 \sqrt{c-\frac{c}{a x}} x^2}{96 a^2}+\frac{17 \sqrt{c-\frac{c}{a x}} x^3}{24 a}-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{\operatorname{Subst}\left (\int \frac{\frac{447 c^4}{8}-\frac{321 c^4 x}{8 a}}{x^2 (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{24 a^2 c^3}\\ &=\frac{149 \sqrt{c-\frac{c}{a x}} x}{64 a^3}-\frac{107 \sqrt{c-\frac{c}{a x}} x^2}{96 a^2}+\frac{17 \sqrt{c-\frac{c}{a x}} x^3}{24 a}-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4+\frac{\operatorname{Subst}\left (\int \frac{\frac{1089 c^5}{16}-\frac{447 c^5 x}{16 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{24 a^3 c^4}\\ &=\frac{149 \sqrt{c-\frac{c}{a x}} x}{64 a^3}-\frac{107 \sqrt{c-\frac{c}{a x}} x^2}{96 a^2}+\frac{17 \sqrt{c-\frac{c}{a x}} x^3}{24 a}-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4+\frac{(363 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{128 a^4}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^4}\\ &=\frac{149 \sqrt{c-\frac{c}{a x}} x}{64 a^3}-\frac{107 \sqrt{c-\frac{c}{a x}} x^2}{96 a^2}+\frac{17 \sqrt{c-\frac{c}{a x}} x^3}{24 a}-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{363 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{64 a^3}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{a^3}\\ &=\frac{149 \sqrt{c-\frac{c}{a x}} x}{64 a^3}-\frac{107 \sqrt{c-\frac{c}{a x}} x^2}{96 a^2}+\frac{17 \sqrt{c-\frac{c}{a x}} x^3}{24 a}-\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{363 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{64 a^4}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.136369, size = 116, normalized size = 0.67 \[ \frac{a x \left (-48 a^3 x^3+136 a^2 x^2-214 a x+447\right ) \sqrt{c-\frac{c}{a x}}-1089 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )+768 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{192 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[c - c/(a*x)]*x*(447 - 214*a*x + 136*a^2*x^2 - 48*a^3*x^3) - 1089*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqr
t[c]] + 768*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(192*a^4)

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Maple [A]  time = 0.117, size = 259, normalized size = 1.5 \begin{align*} -{\frac{x}{384}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 96\,x \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{9/2}\sqrt{{a}^{-1}}-176\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{7/2}\sqrt{{a}^{-1}}+252\,\sqrt{a{x}^{2}-x}{a}^{7/2}\sqrt{{a}^{-1}}x-126\,\sqrt{a{x}^{2}-x}{a}^{5/2}\sqrt{{a}^{-1}}-768\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}+768\,{a}^{3/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ) +1152\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}-63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{a}^{2} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(c*(a*x-1)/a/x)^(1/2)*x*(96*x*(a*x^2-x)^(3/2)*a^(9/2)*(1/a)^(1/2)-176*(a*x^2-x)^(3/2)*a^(7/2)*(1/a)^(1/
2)+252*(a*x^2-x)^(1/2)*a^(7/2)*(1/a)^(1/2)*x-126*(a*x^2-x)^(1/2)*a^(5/2)*(1/a)^(1/2)-768*a^(5/2)*(1/a)^(1/2)*(
(a*x-1)*x)^(1/2)+768*a^(3/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x-1)*x)^(1/2)*a-3*a*x+1)/(a*x+1))+1152*a^2*
ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*(1/a)^(1/2)-63*ln(1/2*(2*(a*x^2-x)^(1/2)*a^(1/2)+2*a*x-1
)/a^(1/2))*(1/a)^(1/2)*a^2)/((a*x-1)*x)^(1/2)/a^(11/2)/(1/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.54578, size = 666, normalized size = 3.87 \begin{align*} \left [\frac{768 \, \sqrt{2} \sqrt{c} \log \left (-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - 2 \,{\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} + 1089 \, \sqrt{c} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{384 \, a^{4}}, -\frac{768 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) +{\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} - 1089 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{192 \, a^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/384*(768*sqrt(2)*sqrt(c)*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) - 2*
(48*a^4*x^4 - 136*a^3*x^3 + 214*a^2*x^2 - 447*a*x)*sqrt((a*c*x - c)/(a*x)) + 1089*sqrt(c)*log(-2*a*c*x + 2*a*s
qrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^4, -1/192*(768*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c
*x - c)/(a*x))/c) + (48*a^4*x^4 - 136*a^3*x^3 + 214*a^2*x^2 - 447*a*x)*sqrt((a*c*x - c)/(a*x)) - 1089*sqrt(-c)
*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^4]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{x^{3} \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx - \int \frac{a x^{4} \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-x**3*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(a*x**4*sqrt(c - c/(a*x))/(a*x + 1), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError