3.1.13 \(\int (c+d x) \tanh ^3(e+f x) \, dx\) [13]
Optimal. Leaf size=100 \[ \frac {d x}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f} \]
[Out]
1/2*d*x/f-1/2*(d*x+c)^2/d+(d*x+c)*ln(1+exp(2*f*x+2*e))/f+1/2*d*polylog(2,-exp(2*f*x+2*e))/f^2-1/2*d*tanh(f*x+e
)/f^2-1/2*(d*x+c)*tanh(f*x+e)^2/f
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Rubi [A]
time = 0.09, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3554, 8,
3799, 2221, 2317, 2438} \begin {gather*} \frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {d \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {d \tanh (e+f x)}{2 f^2}+\frac {d x}{2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x)*Tanh[e + f*x]^3,x]
[Out]
(d*x)/(2*f) - (c + d*x)^2/(2*d) + ((c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (d*PolyLog[2, -E^(2*(e + f*x))])/(2
*f^2) - (d*Tanh[e + f*x])/(2*f^2) - ((c + d*x)*Tanh[e + f*x]^2)/(2*f)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2221
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2317
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Rule 2438
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
Rule 3554
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 3799
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Rule 3801
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
Rubi steps
\begin {align*} \int (c+d x) \tanh ^3(e+f x) \, dx &=-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}+\frac {d \int \tanh ^2(e+f x) \, dx}{2 f}+\int (c+d x) \tanh (e+f x) \, dx\\ &=-\frac {(c+d x)^2}{2 d}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}+2 \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx+\frac {d \int 1 \, dx}{2 f}\\ &=\frac {d x}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}-\frac {d \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {d x}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}-\frac {d \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}\\ &=\frac {d x}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.12, size = 263, normalized size = 2.63 \begin {gather*} \frac {c \log (\cosh (e+f x))}{f}+\frac {d x \text {sech}^2(e+f x)}{2 f}+\frac {d \text {csch}(e) \left (e^{-\tanh ^{-1}(\coth (e))} f^2 x^2-\frac {i \coth (e) \left (-f x \left (-\pi +2 i \tanh ^{-1}(\coth (e))\right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\coth (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\coth (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\coth (e))\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )\right )}{\sqrt {1-\coth ^2(e)}}\right ) \text {sech}(e)}{2 f^2 \sqrt {\text {csch}^2(e) \left (-\cosh ^2(e)+\sinh ^2(e)\right )}}-\frac {d \text {sech}(e) \text {sech}(e+f x) \sinh (f x)}{2 f^2}+\frac {1}{2} d x^2 \tanh (e)-\frac {c \tanh ^2(e+f x)}{2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)*Tanh[e + f*x]^3,x]
[Out]
(c*Log[Cosh[e + f*x]])/f + (d*x*Sech[e + f*x]^2)/(2*f) + (d*Csch[e]*((f^2*x^2)/E^ArcTanh[Coth[e]] - (I*Coth[e]
*(-(f*x*(-Pi + (2*I)*ArcTanh[Coth[e]])) - Pi*Log[1 + E^(2*f*x)] - 2*(I*f*x + I*ArcTanh[Coth[e]])*Log[1 - E^((2
*I)*(I*f*x + I*ArcTanh[Coth[e]]))] + Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Coth[e]]*Log[I*Sinh[f*x + ArcTanh[Coth[
e]]]] + I*PolyLog[2, E^((2*I)*(I*f*x + I*ArcTanh[Coth[e]]))]))/Sqrt[1 - Coth[e]^2])*Sech[e])/(2*f^2*Sqrt[Csch[
e]^2*(-Cosh[e]^2 + Sinh[e]^2)]) - (d*Sech[e]*Sech[e + f*x]*Sinh[f*x])/(2*f^2) + (d*x^2*Tanh[e])/2 - (c*Tanh[e
+ f*x]^2)/(2*f)
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Maple [A]
time = 1.40, size = 166, normalized size = 1.66
| | |
| method |
result |
size |
| | |
| risch |
\(-\frac {d \,x^{2}}{2}+c x +\frac {2 d f x \,{\mathrm e}^{2 f x +2 e}+2 c f \,{\mathrm e}^{2 f x +2 e}+d \,{\mathrm e}^{2 f x +2 e}+d}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+\frac {c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 c \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {2 d e x}{f}-\frac {d \,e^{2}}{f^{2}}+\frac {d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+\frac {2 d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}\) |
\(166\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)*tanh(f*x+e)^3,x,method=_RETURNVERBOSE)
[Out]
-1/2*d*x^2+c*x+(2*d*f*x*exp(2*f*x+2*e)+2*c*f*exp(2*f*x+2*e)+d*exp(2*f*x+2*e)+d)/f^2/(1+exp(2*f*x+2*e))^2+1/f*c
*ln(1+exp(2*f*x+2*e))-2/f*c*ln(exp(f*x+e))-2/f*d*e*x-1/f^2*d*e^2+1/f*d*ln(1+exp(2*f*x+2*e))*x+1/2*d*polylog(2,
-exp(2*f*x+2*e))/f^2+2/f^2*d*e*ln(exp(f*x+e))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="maxima")
[Out]
c*(x + e/f + log(e^(-2*f*x - 2*e) + 1)/f + 2*e^(-2*f*x - 2*e)/(f*(2*e^(-2*f*x - 2*e) + e^(-4*f*x - 4*e) + 1)))
+ 1/2*d*((f^2*x^2*e^(4*f*x + 4*e) + f^2*x^2 + 2*(f^2*x^2*e^(2*e) + 2*f*x*e^(2*e) + e^(2*e))*e^(2*f*x) + 2)/(f
^2*e^(4*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) - 4*integrate(x/(e^(2*f*x + 2*e) + 1), x))
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Fricas [C] Result contains complex when optimal does not.
time = 0.43, size = 2128, normalized size = 21.28 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="fricas")
[Out]
-1/2*(d*f^2*x^2 + (d*f^2*x^2 + 2*c*f^2*x + 4*c*f*cosh(1) - 2*d*cosh(1)^2 - 2*d*sinh(1)^2 + 4*(c*f - d*cosh(1))
*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(d*f^2*x^2 + 2*c*f^2*x + 4*c*f*cosh(1) - 2*d*cosh(1)^2 - 2*d*sin
h(1)^2 + 4*(c*f - d*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (d*f^2*x
^2 + 2*c*f^2*x + 4*c*f*cosh(1) - 2*d*cosh(1)^2 - 2*d*sinh(1)^2 + 4*(c*f - d*cosh(1))*sinh(1))*sinh(f*x + cosh(
1) + sinh(1))^4 + 2*c*f^2*x + 4*c*f*cosh(1) - 2*d*cosh(1)^2 + 2*(d*f^2*x^2 + 4*c*f*cosh(1) - 2*d*cosh(1)^2 - 2
*d*sinh(1)^2 - 2*c*f + 2*(c*f^2 - d*f)*x + 4*(c*f - d*cosh(1))*sinh(1) - d)*cosh(f*x + cosh(1) + sinh(1))^2 -
2*d*sinh(1)^2 + 2*(d*f^2*x^2 + 4*c*f*cosh(1) - 2*d*cosh(1)^2 + 3*(d*f^2*x^2 + 2*c*f^2*x + 4*c*f*cosh(1) - 2*d*
cosh(1)^2 - 2*d*sinh(1)^2 + 4*(c*f - d*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 2*d*sinh(1)^2 - 2*c
*f + 2*(c*f^2 - d*f)*x + 4*(c*f - d*cosh(1))*sinh(1) - d)*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(d*cosh(f*x + co
sh(1) + sinh(1))^4 + 4*d*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + d*sinh(f*x + cosh(1)
+ sinh(1))^4 + 2*d*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(3*d*cosh(f*x + cosh(1) + sinh(1))^2 + d)*sinh(f*x + co
sh(1) + sinh(1))^2 + 4*(d*cosh(f*x + cosh(1) + sinh(1))^3 + d*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1
) + sinh(1)) + d)*dilog(I*cosh(f*x + cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1))) - 2*(d*cosh(f*x + c
osh(1) + sinh(1))^4 + 4*d*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + d*sinh(f*x + cosh(1)
+ sinh(1))^4 + 2*d*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(3*d*cosh(f*x + cosh(1) + sinh(1))^2 + d)*sinh(f*x + c
osh(1) + sinh(1))^2 + 4*(d*cosh(f*x + cosh(1) + sinh(1))^3 + d*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(
1) + sinh(1)) + d)*dilog(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + cosh(1) + sinh(1))) - 2*((c*f - d*cos
h(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(
1))*sinh(f*x + cosh(1) + sinh(1))^3 + (c*f - d*cosh(1) - d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 + 2*(c*f -
d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(3*(c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1
) + sinh(1))^2 + c*f - d*cosh(1) - d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + c*f - d*cosh(1) - d*sinh(1) +
4*((c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^3 + (c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + co
sh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh
(1)) + I) - 2*((c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(c*f - d*cosh(1) - d*sinh(1))
*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (c*f - d*cosh(1) - d*sinh(1))*sinh(f*x + cosh
(1) + sinh(1))^4 + 2*(c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(3*(c*f - d*cosh(1) - d
*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + c*f - d*cosh(1) - d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + c*f
- d*cosh(1) - d*sinh(1) + 4*((c*f - d*cosh(1) - d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^3 + (c*f - d*cosh(1)
- d*sinh(1))*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*log(cosh(f*x + cosh(1) + sinh(1))
+ sinh(f*x + cosh(1) + sinh(1)) - I) - 2*((d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*
(d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (d*f*x + d*cos
h(1) + d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 + d*f*x + 2*(d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(
1) + sinh(1))^2 + 2*(d*f*x + 3*(d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + d*cosh(1) + d
*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + d*cosh(1) + d*sinh(1) + 4*((d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*
x + cosh(1) + sinh(1))^3 + (d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) +
sinh(1)))*log(I*cosh(f*x + cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1)) + 1) - 2*((d*f*x + d*cosh(1)
+ d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))
*sinh(f*x + cosh(1) + sinh(1))^3 + (d*f*x + d*cosh(1) + d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 + d*f*x + 2
*(d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(d*f*x + 3*(d*f*x + d*cosh(1) + d*sinh(1)
)*cosh(f*x + cosh(1) + sinh(1))^2 + d*cosh(1) + d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + d*cosh(1) + d*sin
h(1) + 4*((d*f*x + d*cosh(1) + d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^3 + (d*f*x + d*cosh(1) + d*sinh(1))*co
sh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*log(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x
+ cosh(1) + sinh(1)) + 1) + 4*(c*f - d*cosh(1))*sinh(1) + 4*((d*f^2*x^2 + 2*c*f^2*x + 4*c*f*cosh(1) - 2*d*cosh
(1)^2 - 2*d*sinh(1)^2 + 4*(c*f - d*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^3 + (d*f^2*x^2 + 4*c*f*cosh
(1) - 2*d*cosh(1)^2 - 2*d*sinh(1)^2 - 2*c*f + 2*(c*f^2 - d*f)*x + 4*(c*f - d*cosh(1))*sinh(1) - d)*cosh(f*x +
cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c + d x\right ) \tanh ^{3}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)**3,x)
[Out]
Integral((c + d*x)*tanh(e + f*x)**3, x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="giac")
[Out]
integrate((d*x + c)*tanh(f*x + e)^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tanh}\left (e+f\,x\right )}^3\,\left (c+d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(e + f*x)^3*(c + d*x),x)
[Out]
int(tanh(e + f*x)^3*(c + d*x), x)
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