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]

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

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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 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)]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.85, size = 50, normalized size = 1.43 \[ -\frac {2 \, {\left ({\left (A + 5 \, B\right )} \cosh \relax (x) - {\left (A - B\right )} \sinh \relax (x) + 3 \, A + 3 \, B\right )}}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) + 3\right )}} \]

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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giac [A]  time = 0.12, size = 30, normalized size = 0.86 \[ -\frac {2 \, {\left (3 \, B e^{\left (2 \, x\right )} + 3 \, A e^{x} + 3 \, B e^{x} + A + 2 \, B\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \]

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

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maple [A]  time = 0.04, size = 34, normalized size = 0.97 \[ -\frac {A \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{6}+\frac {B \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{6}+\frac {A \tanh \left (\frac {x}{2}\right )}{2}+\frac {B \tanh \left (\frac {x}{2}\right )}{2} \]

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 = 0.32, size = 129, normalized size = 3.69 \[ \frac {2}{3} \, B {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {2}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1}\right )} + \frac {2}{3} \, A {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {1}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1}\right )} \]

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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mupad [B]  time = 0.08, size = 30, normalized size = 0.86 \[ -\frac {2\,\left (A+2\,B+3\,A\,{\mathrm {e}}^x+3\,B\,{\mathrm {e}}^x+3\,B\,{\mathrm {e}}^{2\,x}\right )}{3\,{\left ({\mathrm {e}}^x+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.61, size = 36, normalized size = 1.03 \[ - \frac {A \tanh ^{3}{\left (\frac {x}{2} \right )}}{6} + \frac {A \tanh {\left (\frac {x}{2} \right )}}{2} + \frac {B \tanh ^{3}{\left (\frac {x}{2} \right )}}{6} + \frac {B \tanh {\left (\frac {x}{2} \right )}}{2} \]

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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