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

Optimal. Leaf size=104 \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{22 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{44 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac{1}{x}\right )^{3/2}} \]

[Out]

(44*(1 + x^(-1))^(3/2)*(1 - x)^(3/2))/(105*(1 - x^(-1))^(3/2)) - (22*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x)/(35*(
1 - x^(-1))^(3/2)) + (2*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x^2)/(7*(1 - x^(-1))^(3/2))

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Rubi [A]  time = 0.126104, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6176, 6181, 78, 45, 37} \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{22 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{44 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac{1}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[x]*(1 - x)^(3/2)*x,x]

[Out]

(44*(1 + x^(-1))^(3/2)*(1 - x)^(3/2))/(105*(1 - x^(-1))^(3/2)) - (22*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x)/(35*(
1 - x^(-1))^(3/2)) + (2*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x^2)/(7*(1 - x^(-1))^(3/2))

Rule 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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]

Rule 78

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] || LtQ
[p, n]))))

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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]

Rubi steps

\begin{align*} \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx &=\frac{(1-x)^{3/2} \int e^{\coth ^{-1}(x)} \left (1-\frac{1}{x}\right )^{3/2} x^{5/2} \, 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 ) \operatorname{Subst}\left (\int \frac{(1-x) \sqrt{1+x}}{x^{9/2}} \, 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^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{\left (11 (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 \left (1-\frac{1}{x}\right )^{3/2}}\\ &=-\frac{22 \left (1+\frac{1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{2 \left (1+\frac{1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{\left (22 (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{35 \left (1-\frac{1}{x}\right )^{3/2}}\\ &=\frac{44 \left (1+\frac{1}{x}\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{22 \left (1+\frac{1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{2 \left (1+\frac{1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0207203, size = 46, normalized size = 0.44 \[ -\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) \left (15 x^2-33 x+22\right )}{105 \sqrt{\frac{x-1}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + x^(-1)]*Sqrt[1 - x]*(1 + x)*(22 - 33*x + 15*x^2))/(105*Sqrt[(-1 + x)/x])

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Maple [A]  time = 0.058, size = 34, normalized size = 0.3 \begin{align*} -{\frac{ \left ( 2+2\,x \right ) \left ( 15\,{x}^{2}-33\,x+22 \right ) }{105}\sqrt{1-x}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(1+x)*(15*x^2-33*x+22)*(1-x)^(1/2)/((-1+x)/(1+x))^(1/2)

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Maxima [C]  time = 1.06617, size = 30, normalized size = 0.29 \begin{align*} -\frac{1}{105} \,{\left (30 i \, x^{3} - 36 i \, x^{2} - 22 i \, x + 44 i\right )} \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(30*I*x^3 - 36*I*x^2 - 22*I*x + 44*I)*sqrt(x + 1)

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Fricas [A]  time = 1.62142, size = 120, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (15 \, x^{4} - 3 \, x^{3} - 29 \, x^{2} + 11 \, x + 22\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{105 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(15*x^4 - 3*x^3 - 29*x^2 + 11*x + 22)*sqrt(-x + 1)*sqrt((x - 1)/(x + 1))/(x - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.2415, size = 78, normalized size = 0.75 \begin{align*} \frac{1}{105} \,{\left (16 i \, \sqrt{2} + \frac{2 \,{\left (15 \,{\left (x + 1\right )}^{3} \sqrt{-x - 1} - 63 \,{\left (x + 1\right )}^{2} \sqrt{-x - 1} - 70 \,{\left (-x - 1\right )}^{\frac{3}{2}}\right )}}{\mathrm{sgn}\left (-x - 1\right )}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(16*I*sqrt(2) + 2*(15*(x + 1)^3*sqrt(-x - 1) - 63*(x + 1)^2*sqrt(-x - 1) - 70*(-x - 1)^(3/2))/sgn(-x - 1
))*sgn(x)