3.5.7 \(\int x \cosh (a+b x) \coth (a+b x) \, dx\) [407]

Optimal. Leaf size=66 \[ -\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \cosh (a+b x)}{b}-\frac {\text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {\text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {\sinh (a+b x)}{b^2} \]

[Out]

-2*x*arctanh(exp(b*x+a))/b+x*cosh(b*x+a)/b-polylog(2,-exp(b*x+a))/b^2+polylog(2,exp(b*x+a))/b^2-sinh(b*x+a)/b^
2

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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5558, 3377, 2717, 4267, 2317, 2438} \begin {gather*} -\frac {\text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {\text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {\sinh (a+b x)}{b^2}+\frac {x \cosh (a+b x)}{b}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

(-2*x*ArcTanh[E^(a + b*x)])/b + (x*Cosh[a + b*x])/b - PolyLog[2, -E^(a + b*x)]/b^2 + PolyLog[2, E^(a + b*x)]/b
^2 - Sinh[a + b*x]/b^2

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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 131, normalized size = 1.98 \begin {gather*} -\frac {-b x \cosh (a+b x)-a \log \left (1-e^{-a-b x}\right )-b x \log \left (1-e^{-a-b x}\right )+a \log \left (1+e^{-a-b x}\right )+b x \log \left (1+e^{-a-b x}\right )+a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )-\text {PolyLog}\left (2,-e^{-a-b x}\right )+\text {PolyLog}\left (2,e^{-a-b x}\right )+\sinh (a+b x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(138\) vs. \(2(63)=126\).
time = 2.29, size = 139, normalized size = 2.11

method	result	size
risch	\(\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}+\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}-\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(139\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(b*x+a)^2*csch(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.33, size = 94, normalized size = 1.42 \begin {gather*} \frac {{\left ({\left (b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b x + 1\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{2}} - \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^{2}} + \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^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)^2*csch(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (61) = 122\).
time = 0.41, size = 255, normalized size = 3.86 \begin {gather*} \frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{2} + b x + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (a \cosh \left (b x + a\right ) + a \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)^2*csch(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)**2*csch(b*x+a),x)

[Out]

Integral(x*cosh(a + b*x)**2*csch(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*cosh(b*x+a)^2*csch(b*x+a),x, algorithm="giac")

[Out]

integrate(x*cosh(b*x + a)^2*csch(b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*cosh(a + b*x)^2)/sinh(a + b*x),x)

[Out]

int((x*cosh(a + b*x)^2)/sinh(a + b*x), x)

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