\(\int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx\) [323]

Optimal result

Integrand size = 13, antiderivative size = 107 \[ \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx=\frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {6 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}} \]

[Out]

12/5*((-1+x)/x)^(1/2)*(1+x)^(1/2)/(1+1/x)^(1/2)+6/5*x*((-1+x)/x)^(1/2)*(1+x)^(1/2)/(1+1/x)^(1/2)+2/5*x^2*((-1+
x)/x)^(1/2)*(1+x)^(1/2)/(1+1/x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6311, 6316, 79, 47, 37} \[ \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx=\frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x^2}{5 \sqrt {\frac {1}{x}+1}}+\frac {6 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{5 \sqrt {\frac {1}{x}+1}}+\frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{5 \sqrt {\frac {1}{x}+1}} \]

[In]

Int[E^ArcCoth[x]*x*Sqrt[1 + x],x]

[Out]

(12*Sqrt[-((1 - x)/x)]*Sqrt[1 + x])/(5*Sqrt[1 + x^(-1)]) + (6*Sqrt[-((1 - x)/x)]*x*Sqrt[1 + x])/(5*Sqrt[1 + x^
(-1)]) + (2*Sqrt[-((1 - x)/x)]*x^2*Sqrt[1 + x])/(5*Sqrt[1 + x^(-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]

Rule 47

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 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 {\sqrt {1+x} \int e^{\coth ^{-1}(x)} \sqrt {1+\frac {1}{x}} x^{3/2} \, dx}{\sqrt {1+\frac {1}{x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1+x}{\sqrt {1-x} x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1+\frac {1}{x}}} \\ & = \frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}-\frac {\left (9 \sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 \sqrt {1+\frac {1}{x}}} \\ & = \frac {6 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}-\frac {\left (6 \sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 \sqrt {1+\frac {1}{x}}} \\ & = \frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {6 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.36 \[ \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} \left (6+3 x+x^2\right )}{5 \sqrt {1+\frac {1}{x}}} \]

[In]

Integrate[E^ArcCoth[x]*x*Sqrt[1 + x],x]

[Out]

(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(6 + 3*x + x^2))/(5*Sqrt[1 + x^(-1)])

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28

method	result	size
gosper	\(\frac {2 \left (x -1\right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\)	\(30\)
default	\(\frac {2 \left (x -1\right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\)	\(30\)
risch	\(\frac {2 \left (x -1\right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\)	\(30\)

[In]

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

[Out]

2/5*(x-1)*(x^2+3*x+6)/((x-1)/(1+x))^(1/2)/(1+x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.24 \[ \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx=\frac {2}{5} \, {\left (x^{2} + 3 \, x + 6\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]

[In]

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

[Out]

2/5*(x^2 + 3*x + 6)*sqrt(x + 1)*sqrt((x - 1)/(x + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.19 \[ \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx=\frac {2 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x - 6\right )}}{5 \, \sqrt {x - 1}} \]

[In]

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

[Out]

2/5*(x^3 + 2*x^2 + 3*x - 6)/sqrt(x - 1)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 4.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.36 \[ \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx=\sqrt {\frac {x-1}{x+1}}\,\left (\frac {6\,x\,\sqrt {x+1}}{5}+\frac {12\,\sqrt {x+1}}{5}+\frac {2\,x^2\,\sqrt {x+1}}{5}\right ) \]

[In]

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

[Out]

((x - 1)/(x + 1))^(1/2)*((6*x*(x + 1)^(1/2))/5 + (12*(x + 1)^(1/2))/5 + (2*x^2*(x + 1)^(1/2))/5)