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-ln(1+cosh(x))/a

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 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])

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 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]

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 = 0.02, 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

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 50, normalized size = 2.94 \[ \frac {2 \, x \cosh \relax (x) + \cosh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (x + \cosh \relax (x)\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 1}{2 \, {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )}} \]

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))

________________________________________________________________________________________

giac [A]  time = 0.13, size = 32, normalized size = 1.88 \[ \frac {x}{a} + \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} - \frac {2 \, \log \left (e^{x} + 1\right )}{a} \]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 27, normalized size = 1.59 \[ -\frac {\ln \left (1+\mathrm {sech}\relax (x )\right )}{a}+\frac {1}{a \,\mathrm {sech}\relax (x )}+\frac {\ln \left (\mathrm {sech}\relax (x )\right )}{a} \]

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))

________________________________________________________________________________________

maxima [B]  time = 0.32, size = 35, normalized size = 2.06 \[ -\frac {x}{a} + \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} - \frac {2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 15, normalized size = 0.88 \[ -\frac {\ln \left (\mathrm {cosh}\relax (x)+1\right )-\mathrm {cosh}\relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(a + a/cosh(x)),x)

[Out]

-(log(cosh(x) + 1) - cosh(x))/a

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sinh {\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]

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

________________________________________________________________________________________