\(\int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx\) [225]

Optimal result

Integrand size = 19, antiderivative size = 18 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\log (a+b \cosh (c+d x))}{b d} \]

[Out]

ln(a+b*cosh(d*x+c))/b/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2747, 31} \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\log (a+b \cosh (c+d x))}{b d} \]

[In]

Int[Sinh[c + d*x]/(a + b*Cosh[c + d*x]),x]

[Out]

Log[a + b*Cosh[c + d*x]]/(b*d)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(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] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (c+d x)\right )}{b d} \\ & = \frac {\log (a+b \cosh (c+d x))}{b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\log (a+b \cosh (c+d x))}{b d} \]

[In]

Integrate[Sinh[c + d*x]/(a + b*Cosh[c + d*x]),x]

[Out]

Log[a + b*Cosh[c + d*x]]/(b*d)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\ln \left (a +b \cosh \left (d x +c \right )\right )}{b d}\)	\(19\)
default	\(\frac {\ln \left (a +b \cosh \left (d x +c \right )\right )}{b d}\)	\(19\)
risch	\(-\frac {x}{b}-\frac {2 c}{b d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}+1\right )}{b d}\)	\(48\)

[In]

int(sinh(d*x+c)/(a+b*cosh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

ln(a+b*cosh(d*x+c))/b/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {d x - \log \left (\frac {2 \, {\left (b \cosh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{b d} \]

[In]

integrate(sinh(d*x+c)/(a+b*cosh(d*x+c)),x, algorithm="fricas")

[Out]

-(d*x - log(2*(b*cosh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))))/(b*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (14) = 28\).

Time = 0.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\begin {cases} \frac {x \sinh {\left (c \right )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\cosh {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {x \sinh {\left (c \right )}}{a + b \cosh {\left (c \right )}} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {a}{b} + \cosh {\left (c + d x \right )} \right )}}{b d} & \text {otherwise} \end {cases} \]

[In]

integrate(sinh(d*x+c)/(a+b*cosh(d*x+c)),x)

[Out]

Piecewise((x*sinh(c)/a, Eq(b, 0) & Eq(d, 0)), (cosh(c + d*x)/(a*d), Eq(b, 0)), (x*sinh(c)/(a + b*cosh(c)), Eq(
d, 0)), (log(a/b + cosh(c + d*x))/(b*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\log \left (b \cosh \left (d x + c\right ) + a\right )}{b d} \]

[In]

integrate(sinh(d*x+c)/(a+b*cosh(d*x+c)),x, algorithm="maxima")

[Out]

log(b*cosh(d*x + c) + a)/(b*d)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\log \left ({\left | b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b d} \]

[In]

integrate(sinh(d*x+c)/(a+b*cosh(d*x+c)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\ln \left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )}{b\,d} \]

[In]

int(sinh(c + d*x)/(a + b*cosh(c + d*x)),x)

[Out]

log(a + b*cosh(c + d*x))/(b*d)