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

Optimal. Leaf size=18 \[ \frac {(A-B) \sinh (x)}{\cosh (x)+1}+B x \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 18, 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 = {2735, 2648} \[ \frac {(A-B) \sinh (x)}{\cosh (x)+1}+B x \]

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 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 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]

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*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 1.28 \[ \frac {\sinh (x) \left (A+B x \coth \left (\frac {x}{2}\right )-B\right )}{\cosh (x)+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 29, normalized size = 1.61 \[ \frac {B x \cosh \relax (x) + B x \sinh \relax (x) + B x - 2 \, A + 2 \, B}{\cosh \relax (x) + \sinh \relax (x) + 1} \]

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)

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giac [A]  time = 0.12, size = 17, normalized size = 0.94 \[ B x - \frac {2 \, {\left (A - B\right )}}{e^{x} + 1} \]

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)

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maple [A]  time = 0.04, size = 34, normalized size = 1.89 \[ A \tanh \left (\frac {x}{2}\right )-B \tanh \left (\frac {x}{2}\right )-B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 26, normalized size = 1.44 \[ B {\left (x - \frac {2}{e^{\left (-x\right )} + 1}\right )} + \frac {2 \, A}{e^{\left (-x\right )} + 1} \]

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)

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mupad [B]  time = 0.05, size = 19, normalized size = 1.06 \[ B\,x-\frac {2\,A-2\,B}{{\mathrm {e}}^x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cosh(x))/(cosh(x) + 1),x)

[Out]

B*x - (2*A - 2*B)/(exp(x) + 1)

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sympy [A]  time = 0.34, size = 15, normalized size = 0.83 \[ A \tanh {\left (\frac {x}{2} \right )} + B x - B \tanh {\left (\frac {x}{2} \right )} \]

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)

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