3.427 \(\int x \coth (a+b x) \text {csch}(a+b x) \, dx\)

Optimal. Leaf size=25 \[ -\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b} \]

[Out]

-arctanh(cosh(b*x+a))/b^2-x*csch(b*x+a)/b

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5419, 3770} \[ -\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x*Coth[a + b*x]*Csch[a + b*x],x]

[Out]

-(ArcTanh[Cosh[a + b*x]]/b^2) - (x*Csch[a + b*x])/b

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 5419

Int[Coth[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.), x_Symbol] :> -Simp[(
x^(m - n + 1)*Csch[a + b*x^n]^p)/(b*n*p), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Csch[a + b*x^n]^p, x],
x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]

Rubi steps

\begin {align*} \int x \coth (a+b x) \text {csch}(a+b x) \, dx &=-\frac {x \text {csch}(a+b x)}{b}+\frac {\int \text {csch}(a+b x) \, dx}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.05, size = 114, normalized size = 4.56 \[ \frac {\log \left (\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b^2}-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b^2}-\frac {x \text {csch}(a)}{b}+\frac {x \text {csch}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right )}{2 b}+\frac {x \text {sech}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*Csch[a])/b) - Log[Cosh[a/2 + (b*x)/2]]/b^2 + Log[Sinh[a/2 + (b*x)/2]]/b^2 + (x*Csch[a/2]*Csch[a/2 + (b*x)
/2]*Sinh[(b*x)/2])/(2*b) + (x*Sech[a/2]*Sech[a/2 + (b*x)/2]*Sinh[(b*x)/2])/(2*b)

________________________________________________________________________________________

fricas [B]  time = 0.51, size = 169, normalized size = 6.76 \[ -\frac {2 \, b x \cosh \left (b x + a\right ) + 2 \, b x \sinh \left (b x + a\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.15, size = 93, normalized size = 3.72 \[ -\frac {2 \, b x e^{\left (b x + a\right )} + e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) - e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - \log \left (e^{\left (b x + a\right )} + 1\right ) + \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.14, size = 54, normalized size = 2.16 \[ -\frac {2 \,{\mathrm e}^{b x +a} x}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.52, size = 64, normalized size = 2.56 \[ -\frac {2 \, x e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 53, normalized size = 2.12 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^4}}{b^2}\right )}{\sqrt {-b^4}}-\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2*atan((exp(b*x)*exp(a)*(-b^4)^(1/2))/b^2))/(-b^4)^(1/2) - (2*x*exp(a + b*x))/(b*(exp(2*a + 2*b*x) - 1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________