\(\int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\) [342]

Optimal result

Integrand size = 23, antiderivative size = 97 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]

[Out]

-2/3*(-a*c*x+c)^(3/2)/a^3/c-2/7*(-a*c*x+c)^(7/2)/a^3/c^3+4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/
2)*c^(1/2)/a^3-4*(-a*c*x+c)^(1/2)/a^3

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6302, 6265, 21, 90, 52, 65, 212} \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {4 \sqrt {c-a c x}}{a^3} \]

[In]

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

[Out]

(-4*Sqrt[c - a*c*x])/a^3 - (2*(c - a*c*x)^(3/2))/(3*a^3*c) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) + (4*Sqrt[2]*Sq
rt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^3

Rule 21

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6265

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, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || 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)} x^2 \sqrt {c-a c x} \, dx \\ & = -\int \frac {x^2 (1-a x) \sqrt {c-a c x}}{1+a x} \, dx \\ & = -\frac {\int \frac {x^2 (c-a c x)^{3/2}}{1+a x} \, dx}{c} \\ & = -\frac {\int \left (\frac {(c-a c x)^{3/2}}{a^2 (1+a x)}-\frac {(c-a c x)^{5/2}}{a^2 c}\right ) \, dx}{c} \\ & = -\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{a^2 c} \\ & = -\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx}{a^2} \\ & = -\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac {(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{a^2} \\ & = -\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {8 \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a^3} \\ & = -\frac {4 \sqrt {c-a c x}}{a^3}-\frac {2 (c-a c x)^{3/2}}{3 a^3 c}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (-52+16 a x-9 a^2 x^2+3 a^3 x^3\right )+84 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{21 a^3} \]

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(-52 + 16*a*x - 9*a^2*x^2 + 3*a^3*x^3) + 84*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2
]*Sqrt[c])])/(21*a^3)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(\frac {84 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+2 \left (3 a^{3} x^{3}-9 a^{2} x^{2}+16 a x -52\right ) \sqrt {-c \left (a x -1\right )}}{21 a^{3}}\)	\(68\)
risch	\(-\frac {2 \left (3 a^{3} x^{3}-9 a^{2} x^{2}+16 a x -52\right ) \left (a x -1\right ) c}{21 a^{3} \sqrt {-c \left (a x -1\right )}}+\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a^{3}}\)	\(74\)
derivativedivides	\(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+2 c^{3} \sqrt {-a c x +c}-2 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c^{3} a^{3}}\)	\(75\)
default	\(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {2 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-4 c^{3} \sqrt {-a c x +c}+4 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{3} c^{3}}\)	\(75\)

[In]

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

[Out]

1/21*(84*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))+2*(3*a^3*x^3-9*a^2*x^2+16*a*x-52)*(-c
*(a*x-1))^(1/2))/a^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.58 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (21 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}, -\frac {2 \, {\left (42 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - {\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt {-a c x + c}\right )}}{21 \, a^{3}}\right ] \]

[In]

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

[Out]

[2/21*(21*sqrt(2)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + (3*a^3*x^3 - 9*a
^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^3, -2/21*(42*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sq
rt(-c)/c) - (3*a^3*x^3 - 9*a^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^3]

Sympy [A] (verification not implemented)

Time = 3.62 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.38 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c^{3} \sqrt {- a c x + c} + \frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {x^{3}}{3} - \frac {x^{2}}{a} + \frac {2 x}{a^{2}} - \frac {2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a^{2}}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*(2*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 2*c**3*sqrt(-a*c*x + c) +
c**2*(-a*c*x + c)**(3/2)/3 + (-a*c*x + c)**(7/2)/7)/(a**3*c**3), Ne(a*c, 0)), (sqrt(c)*(x**3/3 - x**2/a + 2*x
/a**2 - 2*Piecewise((x, Eq(a, 0)), (log(a*x + 1)/a, True))/a**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (21 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 42 \, \sqrt {-a c x + c} c^{3}\right )}}{21 \, a^{3} c^{3}} \]

[In]

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

[Out]

-2/21*(21*sqrt(2)*c^(7/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c))) + 3*
(-a*c*x + c)^(7/2) + 7*(-a*c*x + c)^(3/2)*c^2 + 42*sqrt(-a*c*x + c)*c^3)/(a^3*c^3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{3} \sqrt {-c}} + \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{18} c^{18} - 7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{18} c^{20} - 42 \, \sqrt {-a c x + c} a^{18} c^{21}\right )}}{21 \, a^{21} c^{21}} \]

[In]

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

[Out]

-4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a^3*sqrt(-c)) + 2/21*(3*(a*c*x - c)^3*sqrt(-a*c*x
+ c)*a^18*c^18 - 7*(-a*c*x + c)^(3/2)*a^18*c^20 - 42*sqrt(-a*c*x + c)*a^18*c^21)/(a^21*c^21)

Mupad [B] (verification not implemented)

Time = 4.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {4\,\sqrt {c-a\,c\,x}}{a^3}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3}-\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^3} \]

[In]

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

[Out]

- (4*(c - a*c*x)^(1/2))/a^3 - (2*(c - a*c*x)^(3/2))/(3*a^3*c) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) - (2^(1/2)*c
^(1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*4i)/a^3