3.116 \(\int \frac {a+b \cosh (x)}{(b+a \cosh (x))^2} \, dx\)

Optimal. Leaf size=11 \[ \frac {\sinh (x)}{a \cosh (x)+b} \]

[Out]

sinh(x)/(b+a*cosh(x))

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Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {2754, 8} \[ \frac {\sinh (x)}{a \cosh (x)+b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x])/(b + a*Cosh[x])^2,x]

[Out]

Sinh[x]/(b + a*Cosh[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2754

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 + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rubi steps

\begin {align*} \int \frac {a+b \cosh (x)}{(b+a \cosh (x))^2} \, dx &=\frac {\sinh (x)}{b+a \cosh (x)}+\frac {\int 0 \, dx}{a^2-b^2}\\ &=\frac {\sinh (x)}{b+a \cosh (x)}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 11, normalized size = 1.00 \[ \frac {\sinh (x)}{a \cosh (x)+b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x])/(b + a*Cosh[x])^2,x]

[Out]

Sinh[x]/(b + a*Cosh[x])

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fricas [B]  time = 1.78, size = 54, normalized size = 4.91 \[ -\frac {2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b+a*cosh(x))^2,x, algorithm="fricas")

[Out]

-2*(b*cosh(x) + b*sinh(x) + a)/(a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*(a^2*cosh(x) + a*b)*si
nh(x))

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giac [B]  time = 0.13, size = 26, normalized size = 2.36 \[ -\frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b+a*cosh(x))^2,x, algorithm="giac")

[Out]

-2*(b*e^x + a)/((a*e^(2*x) + 2*b*e^x + a)*a)

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maple [B]  time = 0.06, size = 29, normalized size = 2.64 \[ \frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cosh(x))/(b+a*cosh(x))^2,x)

[Out]

2*tanh(1/2*x)/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b+a*cosh(x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 1.02, size = 51, normalized size = 4.64 \[ -\frac {\frac {2\,{\mathrm {e}}^x\,\left (a\,b^3-a^3\,b\right )}{a\,\left (a\,b^2-a^3\right )}+2}{a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cosh(x))/(b + a*cosh(x))^2,x)

[Out]

-((2*exp(x)*(a*b^3 - a^3*b))/(a*(a*b^2 - a^3)) + 2)/(a + 2*b*exp(x) + a*exp(2*x))

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sympy [B]  time = 146.49, size = 26, normalized size = 2.36 \[ \frac {2 \tanh {\left (\frac {x}{2} \right )}}{a \tanh ^{2}{\left (\frac {x}{2} \right )} + a - b \tanh ^{2}{\left (\frac {x}{2} \right )} + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b+a*cosh(x))**2,x)

[Out]

2*tanh(x/2)/(a*tanh(x/2)**2 + a - b*tanh(x/2)**2 + b)

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