3.120 \(\int \frac{\coth ^3(x)}{a+b \text{csch}(x)} \, dx\)

Optimal. Leaf size=32 \[ \left (\frac{a}{b^2}+\frac{1}{a}\right ) \log (a+b \text{csch}(x))+\frac{\log (\sinh (x))}{a}-\frac{\text{csch}(x)}{b} \]

[Out]

-(Csch[x]/b) + (a^(-1) + a/b^2)*Log[a + b*Csch[x]] + Log[Sinh[x]]/a

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Rubi [A]  time = 0.0721172, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3885, 894} \[ \left (\frac{a}{b^2}+\frac{1}{a}\right ) \log (a+b \text{csch}(x))+\frac{\log (\sinh (x))}{a}-\frac{\text{csch}(x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(a + b*Csch[x]),x]

[Out]

-(Csch[x]/b) + (a^(-1) + a/b^2)*Log[a + b*Csch[x]] + Log[Sinh[x]]/a

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\coth ^3(x)}{a+b \text{csch}(x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-b^2-x^2}{x (a+x)} \, dx,x,b \text{csch}(x)\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1-\frac{b^2}{a x}+\frac{a^2+b^2}{a (a+x)}\right ) \, dx,x,b \text{csch}(x)\right )}{b^2}\\ &=-\frac{\text{csch}(x)}{b}+\left (\frac{1}{a}+\frac{a}{b^2}\right ) \log (a+b \text{csch}(x))+\frac{\log (\sinh (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0502377, size = 37, normalized size = 1.16 \[ \frac{\left (a^2+b^2\right ) \log (a \sinh (x)+b)+a^2 (-\log (\sinh (x)))-a b \text{csch}(x)}{a b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(a + b*Csch[x]),x]

[Out]

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

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Maple [B]  time = 0.03, size = 106, normalized size = 3.3 \begin{align*}{\frac{1}{2\,b}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{a}{{b}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \right ) }+{\frac{1}{a}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \right ) }-{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{a}{{b}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a+b*csch(x)),x)

[Out]

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

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Maxima [B]  time = 1.00749, size = 111, normalized size = 3.47 \begin{align*} \frac{x}{a} + \frac{2 \, e^{\left (-x\right )}}{b e^{\left (-2 \, x\right )} - b} - \frac{a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac{a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*csch(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.66492, size = 543, normalized size = 16.97 \begin{align*} -\frac{b^{2} x \cosh \left (x\right )^{2} + b^{2} x \sinh \left (x\right )^{2} - b^{2} x + 2 \, a b \cosh \left (x\right ) -{\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} - a^{2} - b^{2}\right )} \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (b^{2} x \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} - a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*csch(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{3}{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3/(a+b*csch(x)),x)

[Out]

Integral(coth(x)**3/(a + b*csch(x)), x)

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Giac [B]  time = 1.19312, size = 108, normalized size = 3.38 \begin{align*} -\frac{a \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{b^{2}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a b^{2}} + \frac{a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b}{b^{2}{\left (e^{\left (-x\right )} - e^{x}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*csch(x)),x, algorithm="giac")

[Out]

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