3.97 \(\int \frac{\text{sech}(x)}{a+b \text{csch}(x)} \, dx\)

Optimal. Leaf size=64 \[ -\frac{b \log (a \sinh (x)+b)}{a^2+b^2}+\frac{\log (-\sinh (x)+i)}{2 (b+i a)}-\frac{\log (\sinh (x)+i)}{2 (-b+i a)} \]

[Out]

Log[I - Sinh[x]]/(2*(I*a + b)) - Log[I + Sinh[x]]/(2*(I*a - b)) - (b*Log[b + a*Sinh[x]])/(a^2 + b^2)

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Rubi [A]  time = 0.110849, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3872, 2721, 801} \[ -\frac{b \log (a \sinh (x)+b)}{a^2+b^2}+\frac{\log (-\sinh (x)+i)}{2 (b+i a)}-\frac{\log (\sinh (x)+i)}{2 (-b+i a)} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]/(a + b*Csch[x]),x]

[Out]

Log[I - Sinh[x]]/(2*(I*a + b)) - Log[I + Sinh[x]]/(2*(I*a - b)) - (b*Log[b + a*Sinh[x]])/(a^2 + b^2)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

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

Mathematica [A]  time = 0.0574008, size = 36, normalized size = 0.56 \[ \frac{-b \log (a \sinh (x)+b)+2 a \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+b \log (\cosh (x))}{a^2+b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]/(a + b*Csch[x]),x]

[Out]

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

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Maple [A]  time = 0.026, size = 84, normalized size = 1.3 \begin{align*} -2\,{\frac{b\ln \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \right ) }{2\,{a}^{2}+2\,{b}^{2}}}+2\,{\frac{b\ln \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }{2\,{a}^{2}+2\,{b}^{2}}}+4\,{\frac{a\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{2\,{a}^{2}+2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)/(a+b*csch(x)),x)

[Out]

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

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Maxima [A]  time = 1.47917, size = 89, normalized size = 1.39 \begin{align*} -\frac{2 \, a \arctan \left (e^{\left (-x\right )}\right )}{a^{2} + b^{2}} - \frac{b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{2} + b^{2}} + \frac{b \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{2} + b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73521, size = 177, normalized size = 2.77 \begin{align*} \frac{2 \, a \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - b \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + b \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} + b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*arctan(cosh(x) + sinh(x)) - b*log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x))) + b*log(2*cosh(x)/(cosh(x) - sin
h(x))))/(a^2 + b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(x)/(a + b*csch(x)), x)

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Giac [A]  time = 1.14872, size = 120, normalized size = 1.88 \begin{align*} -\frac{a b \log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{3} + a b^{2}} + \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a}{2 \,{\left (a^{2} + b^{2}\right )}} + \frac{b \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \,{\left (a^{2} + b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*b*log(abs(-a*(e^(-x) - e^x) + 2*b))/(a^3 + a*b^2) + 1/2*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*a/(a^2 +
b^2) + 1/2*b*log((e^(-x) - e^x)^2 + 4)/(a^2 + b^2)