3.1.3 \(\int x \coth (a+b x) \, dx\) [3]

Optimal. Leaf size=45 \[ -\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2} \]

[Out]

-1/2*x^2+x*ln(1-exp(2*b*x+2*a))/b+1/2*polylog(2,exp(2*b*x+2*a))/b^2

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Rubi [A]
time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3797, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Coth[a + b*x],x]

[Out]

-1/2*x^2 + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog[2, E^(2*(a + b*x))]/(2*b^2)

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int x \coth (a+b x) \, dx &=-\frac {x^2}{2}-2 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx\\ &=-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.04 \begin {gather*} -\frac {x^2}{2}+\frac {x \log \left (1-e^{2 a+2 b x}\right )}{b}+\frac {\text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Coth[a + b*x],x]

[Out]

-1/2*x^2 + (x*Log[1 - E^(2*a + 2*b*x)])/b + PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(121\) vs. \(2(41)=82\).
time = 1.71, size = 122, normalized size = 2.71

method	result	size
risch	\(-\frac {x^{2}}{2}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}+\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*coth(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-1/2*x^2-2/b*a*x-a^2/b^2+1/b*ln(1-exp(b*x+a))*x+1/b^2*ln(1-exp(b*x+a))*a+1/b^2*polylog(2,exp(b*x+a))+1/b*ln(ex
p(b*x+a)+1)*x+1/b^2*polylog(2,-exp(b*x+a))+2/b^2*a*ln(exp(b*x+a))-1/b^2*a*ln(exp(b*x+a)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (40) = 80\).
time = 0.28, size = 98, normalized size = 2.18 \begin {gather*} \frac {1}{2} \, x^{2} \coth \left (b x + a\right ) - b {\left (\frac {x^{2}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} + \frac {x^{2}}{b} - \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{3}} - \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(b*x+a),x, algorithm="maxima")

[Out]

1/2*x^2*coth(b*x + a) - b*(x^2/(b*e^(2*b*x + 2*a) - b) + x^2/b - (b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a
)))/b^3 - (b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (40) = 80\).
time = 0.38, size = 112, normalized size = 2.49 \begin {gather*} -\frac {b^{2} x^{2} - 2 \, b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(b^2*x^2 - 2*b*x*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 2*a*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 2*
(b*x + a)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) - 2*dilog(cosh(b*x + a) + sinh(b*x + a)) - 2*dilog(-cosh(b*x
+ a) - sinh(b*x + a)))/b^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \coth {\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(b*x+a),x)

[Out]

Integral(x*coth(a + b*x), 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(x*coth(b*x+a),x, algorithm="giac")

[Out]

integrate(x*coth(b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\mathrm {coth}\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*coth(a + b*x),x)

[Out]

int(x*coth(a + b*x), x)

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