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

Optimal. Leaf size=18 \[ \frac{\log (a+b \cosh (c+d x))}{b d} \]

[Out]

Log[a + b*Cosh[c + d*x]]/(b*d)

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Rubi [A]  time = 0.0318788, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2668, 31} \[ \frac{\log (a+b \cosh (c+d x))}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*Cosh[c + d*x]]/(b*d)

Rule 2668

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]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx &=\frac{\operatorname{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]  time = 0.0370468, size = 18, normalized size = 1. \[ \frac{\log (a+b \cosh (c+d x))}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*Cosh[c + d*x]]/(b*d)

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Maple [A]  time = 0.004, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\cosh \left ( dx+c \right ) \right ) }{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00467, size = 24, normalized size = 1.33 \begin{align*} \frac{\log \left (b \cosh \left (d x + c\right ) + a\right )}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.86296, size = 104, normalized size = 5.78 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 1.03296, size = 41, normalized size = 2.28 \begin{align*} \begin{cases} \frac{x \sinh{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sinh{\left (c \right )}}{a + b \cosh{\left (c \right )}} & \text{for}\: d = 0 \\\frac{\cosh{\left (c + d x \right )}}{a d} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + \cosh{\left (c + d x \right )} \right )}}{b d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Giac [A]  time = 1.32136, size = 42, normalized size = 2.33 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)