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

Optimal. Leaf size=24 \[ -\frac{A \sinh (x)}{1-\cosh (x)}-B \log (1-\cosh (x)) \]

[Out]

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

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Rubi [A]  time = 0.0859135, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4401, 2648, 2667, 31} \[ -\frac{A \sinh (x)}{1-\cosh (x)}-B \log (1-\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Sinh[x])/(1 - Cosh[x]),x]

[Out]

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

Rule 4401

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

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]

Rule 2667

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

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{A+B \sinh (x)}{1-\cosh (x)} \, dx &=\int \left (-\frac{A}{-1+\cosh (x)}-\frac{B \sinh (x)}{-1+\cosh (x)}\right ) \, dx\\ &=-\left (A \int \frac{1}{-1+\cosh (x)} \, dx\right )-B \int \frac{\sinh (x)}{-1+\cosh (x)} \, dx\\ &=-\frac{A \sinh (x)}{1-\cosh (x)}-B \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\cosh (x)\right )\\ &=-B \log (1-\cosh (x))-\frac{A \sinh (x)}{1-\cosh (x)}\\ \end{align*}

Mathematica [A]  time = 0.0441071, size = 19, normalized size = 0.79 \[ A \coth \left (\frac{x}{2}\right )-2 B \log \left (\sinh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

A*Coth[x/2] - 2*B*Log[Sinh[x/2]]

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Maple [A]  time = 0.017, size = 36, normalized size = 1.5 \begin{align*} B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +{A \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-2\,B\ln \left ( \tanh \left ( x/2 \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06148, size = 27, normalized size = 1.12 \begin{align*} -B \log \left (\cosh \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*sinh(x))/(1-cosh(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.79008, size = 167, normalized size = 6.96 \begin{align*} \frac{B x \cosh \left (x\right ) + B x \sinh \left (x\right ) - B x - 2 \,{\left (B \cosh \left (x\right ) + B \sinh \left (x\right ) - B\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, A}{\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*sinh(x))/(1-cosh(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.665453, size = 31, normalized size = 1.29 \begin{align*} \frac{A}{\tanh{\left (\frac{x}{2} \right )}} - B x + 2 B \log{\left (\tanh{\left (\frac{x}{2} \right )} + 1 \right )} - 2 B \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15983, size = 30, normalized size = 1.25 \begin{align*} B x - 2 \, B \log \left ({\left | e^{x} - 1 \right |}\right ) + \frac{2 \, A}{e^{x} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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