\(\int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx\) [322]

Optimal result

Integrand size = 14, antiderivative size = 68 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {14 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}} \]

[Out]

-14/15*(1+1/x)^(3/2)*(1-x)^(3/2)/(1-1/x)^(3/2)+2/5*(1+1/x)^(3/2)*(1-x)^(3/2)*x/(1-1/x)^(3/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6311, 6316, 79, 37} \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\frac {2 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {14 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac {1}{x}\right )^{3/2}} \]

[In]

Int[E^ArcCoth[x]*(1 - x)^(3/2),x]

[Out]

(-14*(1 + x^(-1))^(3/2)*(1 - x)^(3/2))/(15*(1 - x^(-1))^(3/2)) + (2*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x)/(5*(1
- x^(-1))^(3/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

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

Rule 6311

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

Rule 6316

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

Rubi steps \begin{align*} \text {integral}& = \frac {(1-x)^{3/2} \int e^{\coth ^{-1}(x)} \left (1-\frac {1}{x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{x}\right )^{3/2} x^{3/2}} \\ & = -\frac {\left ((1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {(1-x) \sqrt {1+x}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{x}\right )^{3/2}} \\ & = \frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {\left (7 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 \left (1-\frac {1}{x}\right )^{3/2}} \\ & = -\frac {14 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.63 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {2 \sqrt {1+\frac {1}{x}} \sqrt {1-x} \left (-7-4 x+3 x^2\right )}{15 \sqrt {\frac {-1+x}{x}}} \]

[In]

Integrate[E^ArcCoth[x]*(1 - x)^(3/2),x]

[Out]

(-2*Sqrt[1 + x^(-1)]*Sqrt[1 - x]*(-7 - 4*x + 3*x^2))/(15*Sqrt[(-1 + x)/x])

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.43

method	result	size
gosper	\(-\frac {2 \left (1+x \right ) \left (3 x -7\right ) \sqrt {1-x}}{15 \sqrt {\frac {x -1}{1+x}}}\)	\(29\)
default	\(-\frac {2 \left (1+x \right ) \left (3 x -7\right ) \sqrt {1-x}}{15 \sqrt {\frac {x -1}{1+x}}}\)	\(29\)
risch	\(\frac {2 \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right ) \left (3 x^{2}-4 x -7\right )}{15 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}\, \sqrt {-1-x}}\)	\(57\)

[In]

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

[Out]

-2/15*(1+x)*(3*x-7)*(1-x)^(1/2)/((x-1)/(1+x))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, x^{3} - x^{2} - 11 \, x - 7\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{15 \, {\left (x - 1\right )}} \]

[In]

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

[Out]

-2/15*(3*x^3 - x^2 - 11*x - 7)*sqrt(-x + 1)*sqrt((x - 1)/(x + 1))/(x - 1)

Sympy [F]

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

[In]

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

[Out]

Integral((1 - x)**(3/2)/sqrt((x - 1)/(x + 1)), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.25 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\frac {2}{15} \, {\left (-3 i \, x^{2} + 4 i \, x + 7 i\right )} \sqrt {x + 1} \]

[In]

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

[Out]

2/15*(-3*I*x^2 + 4*I*x + 7*I)*sqrt(x + 1)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {16}{15} i \, \sqrt {2} \mathrm {sgn}\left (x + 1\right ) - \frac {2 \, {\left (3 \, {\left (x + 1\right )}^{2} \sqrt {-x - 1} + 10 \, {\left (-x - 1\right )}^{\frac {3}{2}} + 8 i \, \sqrt {2}\right )} \mathrm {sgn}\left (x\right )}{15 \, \mathrm {sgn}\left (x + 1\right )} \]

[In]

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

[Out]

-16/15*I*sqrt(2)*sgn(x + 1) - 2/15*(3*(x + 1)^2*sqrt(-x - 1) + 10*(-x - 1)^(3/2) + 8*I*sqrt(2))*sgn(x)/sgn(x +
1)

Mupad [B] (verification not implemented)

Time = 4.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.44 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\frac {2\,\left (3\,x-7\right )\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2}{15\,\sqrt {1-x}} \]

[In]

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

[Out]

(2*(3*x - 7)*((x - 1)/(x + 1))^(1/2)*(x + 1)^2)/(15*(1 - x)^(1/2))