3.96 \(\int \frac{\cosh (x)}{a+b \text{csch}(x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\sinh (x)}{a}-\frac{b \log (a \sinh (x)+b)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0800283, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac{\sinh (x)}{a}-\frac{b \log (a \sinh (x)+b)}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + b*Csch[x]),x]

[Out]

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

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 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0124498, size = 19, normalized size = 0.95 \[ \frac{a \sinh (x)-b \log (a \sinh (x)+b)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(a + b*Csch[x]),x]

[Out]

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

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Maple [A]  time = 0.022, size = 31, normalized size = 1.6 \begin{align*}{\frac{1}{a{\rm csch} \left (x\right )}}+{\frac{b\ln \left ({\rm csch} \left (x\right ) \right ) }{{a}^{2}}}-{\frac{b\ln \left ( a+b{\rm csch} \left (x\right ) \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01637, size = 65, normalized size = 3.25 \begin{align*} -\frac{b x}{a^{2}} - \frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} - \frac{b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.58189, size = 246, normalized size = 12.3 \begin{align*} \frac{2 \, b x \cosh \left (x\right ) + a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} - 2 \,{\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (b x + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a}{2 \,{\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\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(cosh(x)/(a+b*csch(x)),x)

[Out]

Integral(cosh(x)/(a + b*csch(x)), x)

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Giac [A]  time = 1.16308, size = 53, normalized size = 2.65 \begin{align*} -\frac{e^{\left (-x\right )} - e^{x}}{2 \, a} - \frac{b \log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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