3.9.36 \(\int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx\) [836]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3370, 2833, 8} \begin {gather*} \frac {\sinh (x)}{a \cosh (x)+b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x])/(b^2 + 2*a*b*Cosh[x] + a^2*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 2833

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]

Rule 3370

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + cos[(d_.) + (e_.)*(x_)]^2*(c_.) + (a_))^(n_)*(cos[(d_.) + (e_.)*(x_)]*(B_
.) + (A_)), x_Symbol] :> Dist[1/(4^n*c^n), Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x], x] /; Fr
eeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx &=\left (4 a^2\right ) \int \frac {a+b \cosh (x)}{\left (2 a b+2 a^2 \cosh (x)\right )^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 = 0.04, size = 11, normalized size = 1.00 \begin {gather*} \frac {\sinh (x)}{b+a \cosh (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sinh[x]/(b + a*Cosh[x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(11)=22\).
time = 0.65, size = 29, normalized size = 2.64

method	result	size
risch	\(-\frac {2 \left ({\mathrm e}^{x} b +a \right )}{a \left ({\mathrm e}^{2 x} a +2 \,{\mathrm e}^{x} b +a \right )}\)	\(27\)
default	\(\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b^2+2*a*b*cosh(x)+a^2*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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (11) = 22\).
time = 0.40, size = 54, normalized size = 4.91 \begin {gather*} -\frac {2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b^2+2*a*b*cosh(x)+a^2*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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).
time = 0.41, size = 26, normalized size = 2.36 \begin {gather*} -\frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))/(b^2+2*a*b*cosh(x)+a^2*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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Mupad [B]
time = 1.93, size = 51, normalized size = 4.64 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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