\(\int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx\) [343]

Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^2} \]

[Out]

2/3*(-a*c*x+c)^(3/2)/a^2/c+2/5*(-a*c*x+c)^(5/2)/a^2/c^2-4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2
)*c^(1/2)/a^2+4*(-a*c*x+c)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6265, 21, 81, 52, 65, 212} \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^2}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {4 \sqrt {c-a c x}}{a^2} \]

[In]

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

[Out]

(4*Sqrt[c - a*c*x])/a^2 + (2*(c - a*c*x)^(3/2))/(3*a^2*c) + (2*(c - a*c*x)^(5/2))/(5*a^2*c^2) - (4*Sqrt[2]*Sqr
t[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^2

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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 \sqrt {c-a c x} \, dx \\ & = -\int \frac {x (1-a x) \sqrt {c-a c x}}{1+a x} \, dx \\ & = -\frac {\int \frac {x (c-a c x)^{3/2}}{1+a x} \, dx}{c} \\ & = \frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{a c} \\ & = \frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx}{a} \\ & = \frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac {(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{a} \\ & = \frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac {8 \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a^2} \\ & = \frac {4 \sqrt {c-a c x}}{a^2}+\frac {2 (c-a c x)^{3/2}}{3 a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (38-11 a x+3 a^2 x^2\right )-60 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{15 a^2} \]

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(38 - 11*a*x + 3*a^2*x^2) - 60*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/
(15*a^2)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(\frac {-60 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\left (6 a^{2} x^{2}-22 a x +76\right ) \sqrt {-c \left (a x -1\right )}}{15 a^{2}}\)	\(59\)
risch	\(-\frac {2 \left (3 a^{2} x^{2}-11 a x +38\right ) \left (a x -1\right ) c}{15 a^{2} \sqrt {-c \left (a x -1\right )}}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a^{2}}\)	\(66\)
derivativedivides	\(\frac {\frac {2 \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c^{2} \sqrt {-a c x +c}-4 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{2} c^{2}}\)	\(73\)
default	\(\frac {\frac {2 \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c^{2} \sqrt {-a c x +c}-4 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{2} c^{2}}\)	\(73\)

[In]

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

[Out]

1/15*(-60*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))+(6*a^2*x^2-22*a*x+76)*(-c*(a*x-1))^(
1/2))/a^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (15 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (3 \, a^{2} x^{2} - 11 \, a x + 38\right )} \sqrt {-a c x + c}\right )}}{15 \, a^{2}}, \frac {2 \, {\left (30 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + {\left (3 \, a^{2} x^{2} - 11 \, a x + 38\right )} \sqrt {-a c x + c}\right )}}{15 \, a^{2}}\right ] \]

[In]

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

[Out]

[2/15*(15*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^2*x^2 - 11*
a*x + 38)*sqrt(-a*c*x + c))/a^2, 2/15*(30*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) + (
3*a^2*x^2 - 11*a*x + 38)*sqrt(-a*c*x + c))/a^2]

Sympy [A] (verification not implemented)

Time = 3.50 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 \sqrt {2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c^{2} \sqrt {- a c x + c} + \frac {c \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{a^{2} c^{2}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {x^{2}}{2} - \frac {2 x}{a} + \frac {2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(2*sqrt(2)*c**3*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 2*c**2*sqrt(-a*c*x + c) +
c*(-a*c*x + c)**(3/2)/3 + (-a*c*x + c)**(5/2)/5)/(a**2*c**2), Ne(a*c, 0)), (sqrt(c)*(x**2/2 - 2*x/a + 2*Piecew
ise((x, Eq(a, 0)), (log(a*x + 1)/a, True))/a), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, \sqrt {2} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {5}{2}} + 5 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 30 \, \sqrt {-a c x + c} c^{2}\right )}}{15 \, a^{2} c^{2}} \]

[In]

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

[Out]

2/15*(15*sqrt(2)*c^(5/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c))) + 3*(
-a*c*x + c)^(5/2) + 5*(-a*c*x + c)^(3/2)*c + 30*sqrt(-a*c*x + c)*c^2)/(a^2*c^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c}} + \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{8} c^{8} + 5 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{8} c^{9} + 30 \, \sqrt {-a c x + c} a^{8} c^{10}\right )}}{15 \, a^{10} c^{10}} \]

[In]

integrate(x*(-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^2*sqrt(-c)) + 2/15*(3*(a*c*x - c)^2*sqrt(-a*c*x +
c)*a^8*c^8 + 5*(-a*c*x + c)^(3/2)*a^8*c^9 + 30*sqrt(-a*c*x + c)*a^8*c^10)/(a^10*c^10)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a^2}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^2\,c}+\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^2\,c^2}+\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^2} \]

[In]

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

[Out]

(4*(c - a*c*x)^(1/2))/a^2 + (2*(c - a*c*x)^(3/2))/(3*a^2*c) + (2*(c - a*c*x)^(5/2))/(5*a^2*c^2) + (2^(1/2)*c^(
1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*4i)/a^2