3.97 \(\int \frac{A+B \cosh (x)}{1-\cosh (x)} \, dx\)

Optimal. Leaf size=20 \[ -\frac{(A+B) \sinh (x)}{1-\cosh (x)}-B x \]

[Out]

-(B*x) - ((A + B)*Sinh[x])/(1 - Cosh[x])

________________________________________________________________________________________

Rubi [A]  time = 0.0395878, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2735, 2648} \[ -\frac{(A+B) \sinh (x)}{1-\cosh (x)}-B x \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(1 - Cosh[x]),x]

[Out]

-(B*x) - ((A + B)*Sinh[x])/(1 - Cosh[x])

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{A+B \cosh (x)}{1-\cosh (x)} \, dx &=-B x-(-A-B) \int \frac{1}{1-\cosh (x)} \, dx\\ &=-B x-\frac{(A+B) \sinh (x)}{1-\cosh (x)}\\ \end{align*}

Mathematica [A]  time = 0.0540247, size = 35, normalized size = 1.75 \[ \frac{2 \sinh \left (\frac{x}{2}\right ) \left ((A+B) \cosh \left (\frac{x}{2}\right )-B x \sinh \left (\frac{x}{2}\right )\right )}{\cosh (x)-1} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(1 - Cosh[x]),x]

[Out]

(2*Sinh[x/2]*((A + B)*Cosh[x/2] - B*x*Sinh[x/2]))/(-1 + Cosh[x])

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 37, normalized size = 1.9 \begin{align*} -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{A \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{B \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cosh(x))/(1-cosh(x)),x)

[Out]

-B*ln(tanh(1/2*x)+1)+1/tanh(1/2*x)*A+1/tanh(1/2*x)*B+B*ln(tanh(1/2*x)-1)

________________________________________________________________________________________

Maxima [A]  time = 1.03264, size = 36, normalized size = 1.8 \begin{align*} -B{\left (x + \frac{2}{e^{\left (-x\right )} - 1}\right )} - \frac{2 \, A}{e^{\left (-x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x)),x, algorithm="maxima")

[Out]

-B*(x + 2/(e^(-x) - 1)) - 2*A/(e^(-x) - 1)

________________________________________________________________________________________

Fricas [A]  time = 2.17154, size = 97, normalized size = 4.85 \begin{align*} -\frac{B x \cosh \left (x\right ) + B x \sinh \left (x\right ) - B x - 2 \, A - 2 \, B}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x)),x, algorithm="fricas")

[Out]

-(B*x*cosh(x) + B*x*sinh(x) - B*x - 2*A - 2*B)/(cosh(x) + sinh(x) - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.66039, size = 15, normalized size = 0.75 \begin{align*} \frac{A}{\tanh{\left (\frac{x}{2} \right )}} - B x + \frac{B}{\tanh{\left (\frac{x}{2} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x)),x)

[Out]

A/tanh(x/2) - B*x + B/tanh(x/2)

________________________________________________________________________________________

Giac [A]  time = 1.1627, size = 22, normalized size = 1.1 \begin{align*} -B x + \frac{2 \,{\left (A + B\right )}}{e^{x} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x)),x, algorithm="giac")

[Out]

-B*x + 2*(A + B)/(e^x - 1)