\(\int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx\) [524]

Optimal result

Integrand size = 27, antiderivative size = 172 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=-\frac {149 \sqrt {c-\frac {c}{a x}} x}{64 a^3}+\frac {107 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2}-\frac {17 \sqrt {c-\frac {c}{a x}} x^3}{24 a}+\frac {1}{4} \sqrt {c-\frac {c}{a x}} x^4+\frac {363 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{64 a^4}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^4} \]

[Out]

363/64*arctanh((c-c/a/x)^(1/2)/c^(1/2))*c^(1/2)/a^4-4*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^(
1/2)/a^4-149/64*x*(c-c/a/x)^(1/2)/a^3+107/96*x^2*(c-c/a/x)^(1/2)/a^2-17/24*x^3*(c-c/a/x)^(1/2)/a+1/4*x^4*(c-c/
a/x)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6302, 6268, 25, 528, 457, 100, 156, 162, 65, 214} \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {363 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{64 a^4}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^4}-\frac {149 x \sqrt {c-\frac {c}{a x}}}{64 a^3}+\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{96 a^2}+\frac {1}{4} x^4 \sqrt {c-\frac {c}{a x}}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{24 a} \]

[In]

Int[(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcCoth[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]*
ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^4

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 65

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 100

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 156

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] && ILtQ[m, -1]

Rule 162

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 214

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

Rule 457

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 528

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 6268

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 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(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 \text {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 {\text {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 {\text {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 {\text {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 {\text {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) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a^4}+\frac {(4 c) \text {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 \text {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 \text {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} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{64 a^4}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.67 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {a \sqrt {c-\frac {c}{a x}} x \left (-447+214 a x-136 a^2 x^2+48 a^3 x^3\right )+1089 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-768 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{192 a^4} \]

[In]

Integrate[(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcCoth[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)]/Sq
rt[c]] - 768*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(192*a^4)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.15

method	result	size
risch	\(\frac {\left (48 a^{3} x^{3}-136 a^{2} x^{2}+214 a x -447\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{192 a^{3}}+\frac {\left (\frac {363 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{128 a^{3} \sqrt {a^{2} c}}+\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{a^{4} \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{a x -1}\)	\(197\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-96 x \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}+176 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}-252 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x +768 \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}+126 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}-768 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right )-1152 a^{2} \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}+63 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2}\right )}{384 \sqrt {\left (a x -1\right ) x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}}\)	\(259\)

[In]

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

[Out]

1/192*(48*a^3*x^3-136*a^2*x^2+214*a*x-447)/a^3*x*(c*(a*x-1)/a/x)^(1/2)+(363/128/a^3*ln((-1/2*a*c+a^2*c*x)/(a^2
*c)^(1/2)+(a^2*c*x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)+2/a^4*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+2*2^(1/2)*c^(1/2)
*(a^2*c*(x+1/a)^2-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a)))/(a*x-1)*(c*(a*x-1)/a/x)^(1/2)*(c*(a*x-1)*a*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.58 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\left [\frac {768 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 2 \, {\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 1089 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{384 \, a^{4}}, \frac {768 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 1089 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{192 \, a^{4}}\right ] \]

[In]

integrate(x^3*(c-c/a/x)^(1/2)*(a*x-1)/(a*x+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*sq
rt(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)*a
rctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^4]

Sympy [F]

\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int \frac {x^{3} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}} x^{3}}{a x + 1} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int \frac {x^3\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]

[In]

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

[Out]

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