3.55 \(\int \frac{\sinh (x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\cosh (x)}{a}-\frac{\log (\cosh (x)+1)}{a} \]

[Out]

Cosh[x]/a - Log[1 + Cosh[x]]/a

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Rubi [A]  time = 0.0730798, antiderivative size = 17, 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} \[ \frac{\cosh (x)}{a}-\frac{\log (\cosh (x)+1)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(a + a*Sech[x]),x]

[Out]

Cosh[x]/a - Log[1 + Cosh[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{\sinh (x)}{a+a \text{sech}(x)} \, dx &=-\int \frac{\cosh (x) \sinh (x)}{-a-a \cosh (x)} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{a (-a+x)} \, dx,x,-a \cosh (x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{-a+x} \, dx,x,-a \cosh (x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a-x}\right ) \, dx,x,-a \cosh (x)\right )}{a^2}\\ &=\frac{\cosh (x)}{a}-\frac{\log (1+\cosh (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0180007, size = 16, normalized size = 0.94 \[ \frac{\cosh (x)-2 \log \left (\cosh \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(a + a*Sech[x]),x]

[Out]

(Cosh[x] - 2*Log[Cosh[x/2]])/a

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Maple [A]  time = 0.017, size = 27, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( 1+{\rm sech} \left (x\right ) \right ) }{a}}+{\frac{1}{a{\rm sech} \left (x\right )}}+{\frac{\ln \left ({\rm sech} \left (x\right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(a+a*sech(x)),x)

[Out]

-1/a*ln(1+sech(x))+1/a/sech(x)+1/a*ln(sech(x))

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Maxima [B]  time = 1.12734, size = 47, normalized size = 2.76 \begin{align*} -\frac{x}{a} + \frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} - \frac{2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.40772, size = 200, normalized size = 11.76 \begin{align*} \frac{2 \, x \cosh \left (x\right ) + \cosh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (x + \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x*cosh(x) + cosh(x)^2 - 4*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) + 1) + 2*(x + cosh(x))*sinh(x) + si
nh(x)^2 + 1)/(a*cosh(x) + a*sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sinh{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(a+a*sech(x)),x)

[Out]

Integral(sinh(x)/(sech(x) + 1), x)/a

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Giac [A]  time = 1.15025, size = 43, normalized size = 2.53 \begin{align*} \frac{x}{a} + \frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} - \frac{2 \, \log \left (e^{x} + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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