\(\int e^{-3 \coth ^{-1}(a x)} (c-\frac {c}{a x}) \, dx\) [447]

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {c \csc ^{-1}(a x)}{a}-\frac {4 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

8*c*(a-1/x)/a^2/(1-1/a^2/x^2)^(1/2)+c*(1-1/a^2/x^2)^(1/2)*x+c*arccsc(a*x)/ 
a-4*c*arctanh((1-1/a^2/x^2)^(1/2))/a
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.56 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.12 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {5 a^2 c x^2 \left ((1+a x) \left (\sqrt {1+\frac {1}{a x}} \left (2-3 a x+a^2 x^2\right )+6 a \sqrt {1-\frac {1}{a x}} x \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-2 a \sqrt {1-\frac {1}{a x}} x \arcsin \left (\frac {1}{a x}\right )\right )-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1+\frac {1}{a x}} x^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )+\sqrt {2} c (-1+a x)^3 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )}{5 a^4 \sqrt {1-\frac {1}{a x}} x^3 (1+a x)} \] Input:

Integrate[(c - c/(a*x))/E^(3*ArcCoth[a*x]),x]
 

Output:

(5*a^2*c*x^2*((1 + a*x)*(Sqrt[1 + 1/(a*x)]*(2 - 3*a*x + a^2*x^2) + 6*a*Sqr 
t[1 - 1/(a*x)]*x*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2*a*Sqrt[1 - 1/(a*x)] 
*x*ArcSin[1/(a*x)]) - 4*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^2*Ar 
cTanh[Sqrt[1 - 1/(a^2*x^2)]]) + Sqrt[2]*c*(-1 + a*x)^3*(1 + a*x)*Hypergeom 
etric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2])/(5*a^4*Sqrt[1 - 1/(a*x)]*x^3*(1 
+ a*x))
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6731, 27, 528, 2338, 538, 223, 243, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (c-\frac {c}{a x}\right ) e^{-3 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6731

\(\displaystyle -\frac {\int \frac {c^4 \left (a-\frac {1}{x}\right )^4 x^2}{a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c \int \frac {\left (a-\frac {1}{x}\right )^4 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^4}\)

\(\Big \downarrow \) 528

\(\displaystyle -\frac {c \left (a^2 \int \frac {\left (a^2-\frac {4 a}{x}-\frac {1}{x^2}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

\(\Big \downarrow \) 2338

\(\displaystyle -\frac {c \left (a^2 \left (a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int \frac {\left (4 a+\frac {1}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

\(\Big \downarrow \) 538

\(\displaystyle -\frac {c \left (a^2 \left (-4 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {c \left (a^2 \left (-4 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {c \left (a^2 \left (-2 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c \left (a^2 \left (4 a^3 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a^2 x \sqrt {1-\frac {1}{a^2 x^2}}-a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c \left (a^2 \left (4 a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {8 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^4}\)

Input:

Int[(c - c/(a*x))/E^(3*ArcCoth[a*x]),x]
 

Output:

-((c*((-8*a^2*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a^2*(-(a^2*Sqrt[1 - 1/ 
(a^2*x^2)]*x) - a*ArcSin[1/(a*x)] + 4*a*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])))/ 
a^4)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 223

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 528

Int[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy 
mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b 
*x^2])), x] + Simp[c^2/a   Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) 
^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr 
eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

rule 6731

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(69)=138\).

Time = 0.10 (sec) , antiderivative size = 376, normalized size of antiderivative = 5.01

method	result	size
default	\(-\frac {\left (-4 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-a^{2} \sqrt {a^{2}}\, x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+4 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+4 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-8 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -2 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -2 a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+8 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -4 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}+4 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) c \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\)	\(376\)

Input:

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

Output:

-(-4*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a^2*x^2-(a^2)^(1/2)*(a^2*x^2-1)^( 
1/2)*a^2*x^2-a^2*(a^2)^(1/2)*x^2*arctan(1/(a^2*x^2-1)^(1/2))+4*ln((a^2*x+( 
(a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^3*x^2+4*((a*x-1)*(a*x+1 
))^(3/2)*(a^2)^(1/2)-8*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a*x-2*(a^2)^(1/ 
2)*(a^2*x^2-1)^(1/2)*a*x-2*a*(a^2)^(1/2)*x*arctan(1/(a^2*x^2-1)^(1/2))+8*l 
n((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^2*x-4*((a*x-1 
)*(a*x+1))^(1/2)*(a^2)^(1/2)-(a^2*x^2-1)^(1/2)*(a^2)^(1/2)-arctan(1/(a^2*x 
^2-1)^(1/2))*(a^2)^(1/2)+4*a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2) 
)/(a^2)^(1/2)))/a*c*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/((a*x-1)*(a*x+1))^ 
(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 4 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 4 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a c x + 9 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a} \] Input:

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

Output:

-(2*c*arctan(sqrt((a*x - 1)/(a*x + 1))) + 4*c*log(sqrt((a*x - 1)/(a*x + 1) 
) + 1) - 4*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (a*c*x + 9*c)*sqrt((a*x 
- 1)/(a*x + 1)))/a
 

Sympy [F]

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

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

Output:

c*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral( 
-2*a*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**2*x*sqr 
t(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))/a
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-2 \, a {\left (\frac {c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {2 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {2 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {4 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}}\right )} \] Input:

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

Output:

-2*a*(c*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2/(a*x + 1) - a^2) + c*arct 
an(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 2*c*log(sqrt((a*x - 1)/(a*x + 1)) + 1) 
/a^2 - 2*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 4*c*sqrt((a*x - 1)/(a* 
x + 1))/a^2)
 

Giac [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \] Input:

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

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 13.56 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {8\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}+\frac {c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{a} \] Input:

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

Output:

(2*c*((a*x - 1)/(a*x + 1))^(1/2))/(a - (a*(a*x - 1))/(a*x + 1)) - (2*c*ata 
n(((a*x - 1)/(a*x + 1))^(1/2)))/a + (c*atan(((a*x - 1)/(a*x + 1))^(1/2)*1i 
)*8i)/a + (8*c*((a*x - 1)/(a*x + 1))^(1/2))/a
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.20 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (-4 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a x -4 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right )+4 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a x +4 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right )+2 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +18 \sqrt {a x +1}\, \sqrt {a x -1}-16 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -16 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+17 a x +17\right )}{2 a \left (a x +1\right )} \] Input:

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

Output:

(c*( - 4*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a*x - 4*atan(sqrt(a*x - 1 
) + sqrt(a*x + 1) - 1) + 4*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a*x + 4 
*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1) + 2*sqrt(a*x + 1)*sqrt(a*x - 1)*a 
*x + 18*sqrt(a*x + 1)*sqrt(a*x - 1) - 16*log((sqrt(a*x - 1) + sqrt(a*x + 1 
))/sqrt(2))*a*x - 16*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 17*a*x 
 + 17))/(2*a*(a*x + 1))