3.98 \(\int \frac{A+B \cosh (x)}{(1-\cosh (x))^2} \, dx\)

Optimal. Leaf size=37 \[ -\frac{(A-2 B) \sinh (x)}{3 (1-\cosh (x))}-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2} \]

[Out]

-((A + B)*Sinh[x])/(3*(1 - Cosh[x])^2) - ((A - 2*B)*Sinh[x])/(3*(1 - Cosh[x]))

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Rubi [A]  time = 0.0408369, antiderivative size = 37, 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 = {2750, 2648} \[ -\frac{(A-2 B) \sinh (x)}{3 (1-\cosh (x))}-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(1 - Cosh[x])^2,x]

[Out]

-((A + B)*Sinh[x])/(3*(1 - Cosh[x])^2) - ((A - 2*B)*Sinh[x])/(3*(1 - Cosh[x]))

Rule 2750

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

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))^2} \, dx &=-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2}+\frac{1}{3} (A-2 B) \int \frac{1}{1-\cosh (x)} \, dx\\ &=-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2}-\frac{(A-2 B) \sinh (x)}{3 (1-\cosh (x))}\\ \end{align*}

Mathematica [A]  time = 0.0485829, size = 25, normalized size = 0.68 \[ \frac{\sinh (x) ((A-2 B) \cosh (x)-2 A+B)}{3 (\cosh (x)-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(1 - Cosh[x])^2,x]

[Out]

((-2*A + B + (A - 2*B)*Cosh[x])*Sinh[x])/(3*(-1 + Cosh[x])^2)

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Maple [A]  time = 0.013, size = 26, normalized size = 0.7 \begin{align*} -{\frac{-A+B}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{A+B}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cosh(x))/(1-cosh(x))^2,x)

[Out]

-1/2*(-A+B)/tanh(1/2*x)-1/6*(A+B)/tanh(1/2*x)^3

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Maxima [B]  time = 1.04776, size = 177, normalized size = 4.78 \begin{align*} -\frac{2}{3} \, B{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{2}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1}\right )} + \frac{2}{3} \, A{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{1}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))^2,x, algorithm="maxima")

[Out]

-2/3*B*(3*e^(-x)/(3*e^(-x) - 3*e^(-2*x) + e^(-3*x) - 1) - 3*e^(-2*x)/(3*e^(-x) - 3*e^(-2*x) + e^(-3*x) - 1) -
2/(3*e^(-x) - 3*e^(-2*x) + e^(-3*x) - 1)) + 2/3*A*(3*e^(-x)/(3*e^(-x) - 3*e^(-2*x) + e^(-3*x) - 1) - 1/(3*e^(-
x) - 3*e^(-2*x) + e^(-3*x) - 1))

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Fricas [A]  time = 2.06134, size = 163, normalized size = 4.41 \begin{align*} \frac{2 \,{\left ({\left (A - 5 \, B\right )} \cosh \left (x\right ) -{\left (A + B\right )} \sinh \left (x\right ) - 3 \, A + 3 \, B\right )}}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))^2,x, algorithm="fricas")

[Out]

2/3*((A - 5*B)*cosh(x) - (A + B)*sinh(x) - 3*A + 3*B)/(cosh(x)^2 + 2*(cosh(x) - 1)*sinh(x) + sinh(x)^2 - 4*cos
h(x) + 3)

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Sympy [A]  time = 1.20393, size = 36, normalized size = 0.97 \begin{align*} \frac{A}{2 \tanh{\left (\frac{x}{2} \right )}} - \frac{A}{6 \tanh ^{3}{\left (\frac{x}{2} \right )}} - \frac{B}{2 \tanh{\left (\frac{x}{2} \right )}} - \frac{B}{6 \tanh ^{3}{\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))**2,x)

[Out]

A/(2*tanh(x/2)) - A/(6*tanh(x/2)**3) - B/(2*tanh(x/2)) - B/(6*tanh(x/2)**3)

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Giac [A]  time = 1.16434, size = 43, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (3 \, B e^{\left (2 \, x\right )} + 3 \, A e^{x} - 3 \, B e^{x} - A + 2 \, B\right )}}{3 \,{\left (e^{x} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))^2,x, algorithm="giac")

[Out]

-2/3*(3*B*e^(2*x) + 3*A*e^x - 3*B*e^x - A + 2*B)/(e^x - 1)^3