3.104 \(\int \frac{\text{csch}^4(x)}{a+b \coth (x)} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\left (a^2-b^2\right ) \log (a+b \coth (x))}{b^3}+\frac{a \coth (x)}{b^2}-\frac{\coth ^2(x)}{2 b} \]

[Out]

(a*Coth[x])/b^2 - Coth[x]^2/(2*b) - ((a^2 - b^2)*Log[a + b*Coth[x]])/b^3

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Rubi [A]  time = 0.0674236, antiderivative size = 40, 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 = {3506, 697} \[ -\frac{\left (a^2-b^2\right ) \log (a+b \coth (x))}{b^3}+\frac{a \coth (x)}{b^2}-\frac{\coth ^2(x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(a + b*Coth[x]),x]

[Out]

(a*Coth[x])/b^2 - Coth[x]^2/(2*b) - ((a^2 - b^2)*Log[a + b*Coth[x]])/b^3

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\text{csch}^4(x)}{a+b \coth (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-\frac{x^2}{b^2}}{a+x} \, dx,x,b \coth (x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{b^2}-\frac{x}{b^2}+\frac{-a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \coth (x)\right )}{b}\\ &=\frac{a \coth (x)}{b^2}-\frac{\coth ^2(x)}{2 b}-\frac{\left (a^2-b^2\right ) \log (a+b \coth (x))}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.137139, size = 50, normalized size = 1.25 \[ \frac{2 \left (a^2-b^2\right ) (\log (\sinh (x))-\log (a \sinh (x)+b \cosh (x)))+2 a b \coth (x)-b^2 \text{csch}^2(x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(a + b*Coth[x]),x]

[Out]

(2*a*b*Coth[x] - b^2*Csch[x]^2 + 2*(a^2 - b^2)*(Log[Sinh[x]] - Log[b*Cosh[x] + a*Sinh[x]]))/(2*b^3)

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Maple [B]  time = 0.031, size = 116, normalized size = 2.9 \begin{align*} -{\frac{1}{8\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{a}{2\,{b}^{2}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{8\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{a}{2\,{b}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,a\tanh \left ( x/2 \right ) +b \right ) }+{\frac{1}{b}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,a\tanh \left ( x/2 \right ) +b \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(a+b*coth(x)),x)

[Out]

-1/8/b*tanh(1/2*x)^2+1/2/b^2*a*tanh(1/2*x)-1/8/b/tanh(1/2*x)^2+1/b^3*ln(tanh(1/2*x))*a^2-1/b*ln(tanh(1/2*x))+1
/2*a/b^2/tanh(1/2*x)-1/b^3*ln(tanh(1/2*x)^2*b+2*a*tanh(1/2*x)+b)*a^2+1/b*ln(tanh(1/2*x)^2*b+2*a*tanh(1/2*x)+b)

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Maxima [B]  time = 1.0331, size = 149, normalized size = 3.72 \begin{align*} \frac{2 \,{\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} - a\right )}}{2 \, b^{2} e^{\left (-2 \, x\right )} - b^{2} e^{\left (-4 \, x\right )} - b^{2}} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{b^{3}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{3}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a+b*coth(x)),x, algorithm="maxima")

[Out]

2*((a + b)*e^(-2*x) - a)/(2*b^2*e^(-2*x) - b^2*e^(-4*x) - b^2) - (a^2 - b^2)*log(-(a - b)*e^(-2*x) + a + b)/b^
3 + (a^2 - b^2)*log(e^(-x) + 1)/b^3 + (a^2 - b^2)*log(e^(-x) - 1)/b^3

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Fricas [B]  time = 2.72258, size = 1087, normalized size = 27.18 \begin{align*} \frac{2 \,{\left (a b - b^{2}\right )} \cosh \left (x\right )^{2} + 4 \,{\left (a b - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 2 \,{\left (a b - b^{2}\right )} \sinh \left (x\right )^{2} - 2 \, a b -{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} - 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \,{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} -{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} - 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \,{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} -{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{b^{3} \cosh \left (x\right )^{4} + 4 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{3} \sinh \left (x\right )^{4} - 2 \, b^{3} \cosh \left (x\right )^{2} + b^{3} + 2 \,{\left (3 \, b^{3} \cosh \left (x\right )^{2} - b^{3}\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b^{3} \cosh \left (x\right )^{3} - b^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a+b*coth(x)),x, algorithm="fricas")

[Out]

(2*(a*b - b^2)*cosh(x)^2 + 4*(a*b - b^2)*cosh(x)*sinh(x) + 2*(a*b - b^2)*sinh(x)^2 - 2*a*b - ((a^2 - b^2)*cosh
(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 - 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*c
osh(x)^2 - a^2 + b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 - (a^2 - b^2)*cosh(x))*sinh(x))*log(2*(
b*cosh(x) + a*sinh(x))/(cosh(x) - sinh(x))) + ((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a^2
- b^2)*sinh(x)^4 - 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*cosh(x)^2 - a^2 + b^2)*sinh(x)^2 + a^2 - b^2 + 4
*((a^2 - b^2)*cosh(x)^3 - (a^2 - b^2)*cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))))/(b^3*cosh(x)^4 + 4
*b^3*cosh(x)*sinh(x)^3 + b^3*sinh(x)^4 - 2*b^3*cosh(x)^2 + b^3 + 2*(3*b^3*cosh(x)^2 - b^3)*sinh(x)^2 + 4*(b^3*
cosh(x)^3 - b^3*cosh(x))*sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{4}{\left (x \right )}}{a + b \coth{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**4/(a+b*coth(x)),x)

[Out]

Integral(csch(x)**4/(a + b*coth(x)), x)

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Giac [B]  time = 1.17598, size = 143, normalized size = 3.58 \begin{align*} -\frac{{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a b^{3} + b^{4}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b^{3}} - \frac{2 \,{\left (a b -{\left (a b - b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{b^{3}{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a+b*coth(x)),x, algorithm="giac")

[Out]

-(a^3 + a^2*b - a*b^2 - b^3)*log(abs(a*e^(2*x) + b*e^(2*x) - a + b))/(a*b^3 + b^4) + (a^2 - b^2)*log(abs(e^(2*
x) - 1))/b^3 - 2*(a*b - (a*b - b^2)*e^(2*x))/(b^3*(e^(2*x) - 1)^2)