3.2.4 \(\int \frac {\text {csch}^4(x)}{a+b \coth (x)} \, dx\) [104]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {3587, 711} \begin {gather*} -\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} \end {gather*}

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 711

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]

Rule 3587

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]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 50, normalized size = 1.25 \begin {gather*} \frac {2 a b \coth (x)-b^2 \text {csch}^2(x)+2 \left (a^2-b^2\right ) (\log (\sinh (x))-\log (b \cosh (x)+a \sinh (x)))}{2 b^3} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(101\) vs. \(2(38)=76\).
time = 0.71, size = 102, normalized size = 2.55

method	result	size
default	\(\frac {-\frac {b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}+2 a \tanh \left (\frac {x}{2}\right )}{4 b^{2}}-\frac {1}{8 b \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 b^{3}}+\frac {a}{2 b^{2} \tanh \left (\frac {x}{2}\right )}+\frac {\left (-4 a^{2}+4 b^{2}\right ) \ln \left (b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{4 b^{3}}\)	\(102\)
risch	\(\frac {2 \,{\mathrm e}^{2 x} a -2 b \,{\mathrm e}^{2 x}-2 a}{\left ({\mathrm e}^{2 x}-1\right )^{2} b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) a^{2}}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{b}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/b^2*(-1/2*b*tanh(1/2*x)^2+2*a*tanh(1/2*x))-1/8/b/tanh(1/2*x)^2+1/4/b^3*(4*a^2-4*b^2)*ln(tanh(1/2*x))+1/2*a
/b^2/tanh(1/2*x)+1/4/b^3*(-4*a^2+4*b^2)*ln(b*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (38) = 76\).
time = 0.26, size = 110, normalized size = 2.75 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (38) = 76\).
time = 0.39, size = 434, normalized size = 10.85 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{4}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (38) = 76\).
time = 0.43, size = 106, normalized size = 2.65 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 1.44, size = 88, normalized size = 2.20 \begin {gather*} \frac {2\,\left (a-b\right )}{b^2\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2}{b\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3}+\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(a + b*coth(x))),x)

[Out]

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

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