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

Optimal. Leaf size=12 \[ -\frac{\cosh (x)}{a \sinh (x)+b} \]

[Out]

-(Cosh[x]/(b + a*Sinh[x]))

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Rubi [A]  time = 0.0321646, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2754, 8} \[ -\frac{\cosh (x)}{a \sinh (x)+b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[x]/(b + a*Sinh[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]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.03527, size = 12, normalized size = 1. \[ -\frac{\cosh (x)}{a \sinh (x)+b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[x]/(b + a*Sinh[x]))

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Maple [B]  time = 0.029, size = 36, normalized size = 3. \begin{align*} -2\,{\frac{1}{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b} \left ( -{\frac{a\tanh \left ( x/2 \right ) }{b}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.63292, size = 158, normalized size = 13.17 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-b*sinh(x))/(b+a*sinh(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)*sin
h(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.23304, size = 41, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (b e^{x} - a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} - a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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