\(\int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx\) [77]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.08 (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 = {2881, 12, 2826, 3855, 2739, 632, 212} \[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=-\frac {2 b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {b \text {arctanh}(\cosh (x))}{a^2}-\frac {\coth (x)}{a} \]

[In]

Int[Csch[x]^2/(a + b*Sinh[x]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2826

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[b/(
b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; Fre
eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2
- b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])
^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m +
n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||
!(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

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

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.53 \[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=-\frac {a \coth \left (\frac {x}{2}\right )+2 b \left (-\frac {2 b \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+a \tanh \left (\frac {x}{2}\right )}{2 a^2} \]

[In]

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

[Out]

-1/2*(a*Coth[x/2] + 2*b*((-2*b*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - Log[Cosh[x/2]] +
Log[Sinh[x/2]]) + a*Tanh[x/2])/a^2

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24

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

[In]

int(csch(x)^2/(a+b*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

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

Time = 0.29 (sec) , antiderivative size = 345, normalized size of antiderivative = 5.85 \[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=\frac {2 \, a^{3} + 2 \, a b^{2} - {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) + {\left (a^{2} b + b^{3} - {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{2} b + b^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a^{2} b + b^{3} - {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{2} b + b^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{4} + a^{2} b^{2} - {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (x\right )^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {csch}^{2}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69 \[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=\frac {b^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{2}} + \frac {b \log \left (e^{\left (-x\right )} + 1\right )}{a^{2}} - \frac {b \log \left (e^{\left (-x\right )} - 1\right )}{a^{2}} + \frac {2}{a e^{\left (-2 \, x\right )} - a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.66 \[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=\frac {b^{2} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{2}} + \frac {b \log \left (e^{x} + 1\right )}{a^{2}} - \frac {b \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{2}} - \frac {2}{a {\left (e^{\left (2 \, x\right )} - 1\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 292, normalized size of antiderivative = 4.95 \[ \int \frac {\text {csch}^2(x)}{a+b \sinh (x)} \, dx=\frac {2}{a-a\,{\mathrm {e}}^{2\,x}}-\frac {b\,\ln \left (32\,{\mathrm {e}}^x-32\right )}{a^2}+\frac {b\,\ln \left (32\,{\mathrm {e}}^x+32\right )}{a^2}+\frac {b^2\,\ln \left (128\,a^4\,{\mathrm {e}}^x-64\,a\,b^3-64\,a^3\,b-32\,b^3\,\sqrt {a^2+b^2}+32\,b^4\,{\mathrm {e}}^x+128\,a^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x-64\,a^2\,b\,\sqrt {a^2+b^2}+96\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^4+a^2\,b^2}-\frac {b^2\,\ln \left (32\,b^3\,\sqrt {a^2+b^2}-64\,a\,b^3-64\,a^3\,b+128\,a^4\,{\mathrm {e}}^x+32\,b^4\,{\mathrm {e}}^x-128\,a^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x+64\,a^2\,b\,\sqrt {a^2+b^2}-96\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^4+a^2\,b^2} \]

[In]

int(1/(sinh(x)^2*(a + b*sinh(x))),x)

[Out]

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