3.81 \(\int \frac {\text {csch}^2(x)}{a+b \text {csch}(x)} \, dx\)

Optimal. Leaf size=50 \[ \frac {2 a \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3789, 3770, 3831, 2660, 618, 206} \[ \frac {2 a \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + b*Csch[x]),x]

[Out]

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

Rule 206

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

Rule 618

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 2660

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 3770

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

Rule 3789

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 3831

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 58, normalized size = 1.16 \[ \frac {\log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {2 a \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^2/(a + b*Csch[x]),x]

[Out]

((-2*a*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + Log[Tanh[x/2]])/b

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fricas [B]  time = 0.44, size = 156, normalized size = 3.12 \[ \frac {\sqrt {a^{2} + b^{2}} a \log \left (\frac {a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}}{a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, b \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + b\right )} \sinh \relax (x) - a}\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (a^{2} + b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{a^{2} b + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(a^2 + b^2)*a*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)*si
nh(x) + 2*sqrt(a^2 + b^2)*(a*cosh(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^2 + b^2)*log(cosh(x) + sinh(x) + 1) + (a^2 + b^2)*log(cosh(x) + sinh(x) - 1))/(a^2*b
+ b^3)

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giac [A]  time = 0.15, size = 82, normalized size = 1.64 \[ -\frac {a \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b} - \frac {\log \left (e^{x} + 1\right )}{b} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*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) - lo
g(e^x + 1)/b + log(abs(e^x - 1))/b

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maple [A]  time = 0.09, size = 49, normalized size = 0.98 \[ -\frac {2 a \arctanh \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 83, normalized size = 1.66 \[ -\frac {a \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.68, size = 287, normalized size = 5.74 \[ \frac {\ln \left (32\,b-32\,b\,{\mathrm {e}}^x\right )}{b}-\frac {\ln \left (32\,b+32\,b\,{\mathrm {e}}^x\right )}{b}+\frac {a\,\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4-128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x+64\,a\,b^3\,\sqrt {a^2+b^2}+32\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,b+b^3}-\frac {a\,\ln \left (64\,a\,b^4+64\,a^3\,b^2-128\,b^5\,{\mathrm {e}}^x-128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-32\,a^4\,b\,{\mathrm {e}}^x-160\,a^2\,b^3\,{\mathrm {e}}^x+64\,a\,b^3\,\sqrt {a^2+b^2}+32\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,b+b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(a + b/sinh(x))),x)

[Out]

log(32*b - 32*b*exp(x))/b - log(32*b + 32*b*exp(x))/b + (a*log(128*b^5*exp(x) - 64*a^3*b^2 - 64*a*b^4 - 128*b^
4*exp(x)*(a^2 + b^2)^(1/2) + 32*a^4*b*exp(x) + 160*a^2*b^3*exp(x) + 64*a*b^3*(a^2 + b^2)^(1/2) + 32*a^3*b*(a^2
+ b^2)^(1/2) - 96*a^2*b^2*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^2*b + b^3) - (a*log(64*a*b^4 + 64*a
^3*b^2 - 128*b^5*exp(x) - 128*b^4*exp(x)*(a^2 + b^2)^(1/2) - 32*a^4*b*exp(x) - 160*a^2*b^3*exp(x) + 64*a*b^3*(
a^2 + b^2)^(1/2) + 32*a^3*b*(a^2 + b^2)^(1/2) - 96*a^2*b^2*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^2*b
+ b^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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