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

Optimal. Leaf size=35 \[ \frac{(A+2 B) \sinh (x)}{3 (\cosh (x)+1)}+\frac{(A-B) \sinh (x)}{3 (\cosh (x)+1)^2} \]

[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 = 0.0375198, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2750, 2648} \[ \frac{(A+2 B) \sinh (x)}{3 (\cosh (x)+1)}+\frac{(A-B) \sinh (x)}{3 (\cosh (x)+1)^2} \]

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 = 0.0504652, size = 25, normalized size = 0.71 \[ \frac{\sinh (x) ((A+2 B) \cosh (x)+2 A+B)}{3 (\cosh (x)+1)^2} \]

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 = 0.007, size = 34, normalized size = 1. \begin{align*} -{\frac{A}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{B}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{A}{2}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{B}{2}\tanh \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03681, size = 174, normalized size = 4.97 \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 = 2.07598, size = 165, normalized size = 4.71 \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*co
sh(x) + 3)

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Sympy [A]  time = 0.856958, size = 36, normalized size = 1.03 \begin{align*} - \frac{A \tanh ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{A \tanh{\left (\frac{x}{2} \right )}}{2} + \frac{B \tanh ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{B \tanh{\left (\frac{x}{2} \right )}}{2} \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*tanh(x/2)**3/6 + A*tanh(x/2)/2 + B*tanh(x/2)**3/6 + B*tanh(x/2)/2

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Giac [A]  time = 1.21306, size = 41, normalized size = 1.17 \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