\(\int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx\) [82]

Optimal result

Integrand size = 13, antiderivative size = 59 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {2 a^2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}-\frac {\coth (x)}{b} \]

[Out]

a*arctanh(cosh(x))/b^2-coth(x)/b-2*a^2*arctanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/b^2/(a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3875, 3874, 3855, 3916, 2739, 632, 212} \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 a^2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {\coth (x)}{b} \]

[In]

Int[Csch[x]^3/(a + b*Csch[x]),x]

[Out]

(a*ArcTanh[Cosh[x]])/b^2 - (2*a^2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Coth[x]/
b

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3855

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

Rule 3874

Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[Csc[e + f*x],
x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3875

Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-Cot[e + f*x]/(b*f), x
] - Dist[a/b, Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\coth (x)}{b}-\frac {a \int \frac {\text {csch}^2(x)}{a+b \text {csch}(x)} \, dx}{b} \\ & = -\frac {\coth (x)}{b}-\frac {a \int \text {csch}(x) \, dx}{b^2}+\frac {a^2 \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx}{b^2} \\ & = \frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {\coth (x)}{b}+\frac {a^2 \int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx}{b^3} \\ & = \frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {\coth (x)}{b}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^3} \\ & = \frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {\coth (x)}{b}-\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{b^3} \\ & = \frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {2 a^2 \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}-\frac {\coth (x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.56 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\text {csch}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \left (-b \cosh (x)+a \left (\frac {2 a \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x)\right )}{2 b^2} \]

[In]

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

[Out]

(Csch[x/2]*Sech[x/2]*(-(b*Cosh[x]) + a*((2*a*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + Lo
g[Cosh[x/2]] - Log[Sinh[x/2]])*Sinh[x]))/(2*b^2)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24

method	result	size
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2 b}-\frac {2 a^{2} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{2} \sqrt {a^{2}+b^{2}}}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}}\)	\(73\)
risch	\(-\frac {2}{b \left ({\mathrm e}^{2 x}-1\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, b^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, b^{2}}+\frac {a \ln \left ({\mathrm e}^{x}+1\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{x}-1\right )}{b^{2}}\)	\(143\)

[In]

int(csch(x)^3/(a+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (55) = 110\).

Time = 0.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 5.85 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \, a^{2} b + 2 \, b^{3} - {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) + {\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}} \]

[In]

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

[Out]

(2*a^2*b + 2*b^3 - (a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 - a^2)*sqrt(a^2 + b^2)*log((a^2*cosh
(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(a*cos
h(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) - a)) + (a^3 + a*b
^2 - (a^3 + a*b^2)*cosh(x)^2 - 2*(a^3 + a*b^2)*cosh(x)*sinh(x) - (a^3 + a*b^2)*sinh(x)^2)*log(cosh(x) + sinh(x
) + 1) - (a^3 + a*b^2 - (a^3 + a*b^2)*cosh(x)^2 - 2*(a^3 + a*b^2)*cosh(x)*sinh(x) - (a^3 + a*b^2)*sinh(x)^2)*l
og(cosh(x) + sinh(x) - 1))/(a^2*b^2 + b^4 - (a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^2*b^2 + b^4)*cosh(x)*sinh(x) - (a
^2*b^2 + b^4)*sinh(x)^2)

Sympy [F]

\[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a^{2} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{2}} + \frac {a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac {a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac {2}{b e^{\left (-2 \, x\right )} - b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.66 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a^{2} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{2}} + \frac {a \log \left (e^{x} + 1\right )}{b^{2}} - \frac {a \log \left ({\left | e^{x} - 1 \right |}\right )}{b^{2}} - \frac {2}{b {\left (e^{\left (2 \, x\right )} - 1\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.50 (sec) , antiderivative size = 292, normalized size of antiderivative = 4.95 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {2}{b-b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (32\,{\mathrm {e}}^x-32\right )}{b^2}+\frac {a\,\ln \left (32\,{\mathrm {e}}^x+32\right )}{b^2}+\frac {a^2\,\ln \left (32\,a^4\,{\mathrm {e}}^x-64\,a\,b^3-64\,a^3\,b-32\,a^3\,\sqrt {a^2+b^2}+128\,b^4\,{\mathrm {e}}^x+128\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x-64\,a\,b^2\,\sqrt {a^2+b^2}+96\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,b^2+b^4}-\frac {a^2\,\ln \left (32\,a^3\,\sqrt {a^2+b^2}-64\,a\,b^3-64\,a^3\,b+32\,a^4\,{\mathrm {e}}^x+128\,b^4\,{\mathrm {e}}^x-128\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x+64\,a\,b^2\,\sqrt {a^2+b^2}-96\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,b^2+b^4} \]

[In]

int(1/(sinh(x)^3*(a + b/sinh(x))),x)

[Out]

2/(b - b*exp(2*x)) - (a*log(32*exp(x) - 32))/b^2 + (a*log(32*exp(x) + 32))/b^2 + (a^2*log(32*a^4*exp(x) - 64*a
*b^3 - 64*a^3*b - 32*a^3*(a^2 + b^2)^(1/2) + 128*b^4*exp(x) + 128*b^3*exp(x)*(a^2 + b^2)^(1/2) + 160*a^2*b^2*e
xp(x) - 64*a*b^2*(a^2 + b^2)^(1/2) + 96*a^2*b*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(b^4 + a^2*b^2) - (
a^2*log(32*a^3*(a^2 + b^2)^(1/2) - 64*a*b^3 - 64*a^3*b + 32*a^4*exp(x) + 128*b^4*exp(x) - 128*b^3*exp(x)*(a^2
+ b^2)^(1/2) + 160*a^2*b^2*exp(x) + 64*a*b^2*(a^2 + b^2)^(1/2) - 96*a^2*b*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2
)^(1/2))/(b^4 + a^2*b^2)