3.1.76 \(\int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx\) [76]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2826, 3855, 2739, 632, 212} \begin {gather*} \frac {2 b \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(a + b*Sinh[x]),x]

[Out]

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

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 3855

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 58, normalized size = 1.16 \begin {gather*} \frac {-\frac {2 b \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\log \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(a + b*Sinh[x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.44, size = 49, normalized size = 0.98

method	result	size
default	\(-\frac {2 b \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\)	\(49\)
risch	\(\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{a}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{a}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 83, normalized size = 1.66 \begin {gather*} -\frac {b \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} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (46) = 92\).
time = 0.37, size = 156, normalized size = 3.12 \begin {gather*} \frac {\sqrt {a^{2} + b^{2}} b \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^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{3} + a b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)/(a + b*sinh(x)), x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 82, normalized size = 1.64 \begin {gather*} -\frac {b \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} - \frac {\log \left (e^{x} + 1\right )}{a} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.64, size = 287, normalized size = 5.74 \begin {gather*} \frac {\ln \left (32\,a-32\,a\,{\mathrm {e}}^x\right )}{a}-\frac {\ln \left (32\,a+32\,a\,{\mathrm {e}}^x\right )}{a}+\frac {b\,\ln \left (128\,a^5\,{\mathrm {e}}^x-64\,a^2\,b^3-64\,a^4\,b-128\,a^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a\,b^4\,{\mathrm {e}}^x+160\,a^3\,b^2\,{\mathrm {e}}^x+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,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^3+a\,b^2}-\frac {b\,\ln \left (64\,a^4\,b+64\,a^2\,b^3-128\,a^5\,{\mathrm {e}}^x-128\,a^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-32\,a\,b^4\,{\mathrm {e}}^x-160\,a^3\,b^2\,{\mathrm {e}}^x+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,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^3+a\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)*(a + b*sinh(x))),x)

[Out]

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

________________________________________________________________________________________