3.586 \(\int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx\)

Optimal. Leaf size=17 \[ \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3075} \[ \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x] + b*Sinh[x])^(-2),x]

[Out]

Sinh[x]/(a*(a*Cosh[x] + b*Sinh[x]))

Rule 3075

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-2), x_Symbol] :> Simp[Sin[c + d*x]/(a*d*
(a*Cos[c + d*x] + b*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 17, normalized size = 1.00 \[ \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sinh[x]/(a*(a*Cosh[x] + b*Sinh[x]))

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fricas [B]  time = 0.40, size = 62, normalized size = 3.65 \[ -\frac {2}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)+b*sinh(x))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.12, size = 26, normalized size = 1.53 \[ -\frac {2}{{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )} {\left (a + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)+b*sinh(x))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.24, size = 29, normalized size = 1.71 \[ \frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)+b*sinh(x))^2,x)

[Out]

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

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maxima [A]  time = 0.36, size = 29, normalized size = 1.71 \[ \frac {2}{a^{2} - b^{2} + {\left (a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)+b*sinh(x))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.54, size = 22, normalized size = 1.29 \[ -\frac {2}{\left (a+b\right )\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x) + b*sinh(x))^2,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)+b*sinh(x))**2,x)

[Out]

Timed out

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