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

Optimal. Leaf size=13 \[ -\frac{1}{b (a+b \sinh (x))} \]

[Out]

-(1/(b*(a + b*Sinh[x])))

________________________________________________________________________________________

Rubi [A]  time = 0.0253128, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2668, 32} \[ -\frac{1}{b (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + b*Sinh[x])^2,x]

[Out]

-(1/(b*(a + b*Sinh[x])))

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0131484, size = 13, normalized size = 1. \[ -\frac{1}{b (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*(a + b*Sinh[x])))

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b \left ( a+b\sinh \left ( x \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b/(a+b*sinh(x))

________________________________________________________________________________________

Maxima [A]  time = 1.2087, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{{\left (b \sinh \left (x\right ) + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*sinh(x) + a)*b)

________________________________________________________________________________________

Fricas [B]  time = 1.95935, size = 149, normalized size = 11.46 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.01477, size = 32, normalized size = 2.46 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sinh{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\tilde{\infty } \sinh{\left (x \right )} & \text{for}\: a = - b \sinh{\left (x \right )} \\\frac{\sinh{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{1}{a b + b^{2} \sinh{\left (x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/sinh(x), Eq(a, 0) & Eq(b, 0)), (zoo*sinh(x), Eq(a, -b*sinh(x))), (sinh(x)/a**2, Eq(b, 0)), (-1/
(a*b + b**2*sinh(x)), True))

________________________________________________________________________________________

Giac [A]  time = 1.12229, size = 30, normalized size = 2.31 \begin{align*} \frac{2}{{\left (b{\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((b*(e^(-x) - e^x) - 2*a)*b)