3.1055 \(\int \frac {\cosh (a+b x)-\sinh (a+b x)}{\cosh (a+b x)+\sinh (a+b x)} \, dx\)

Optimal. Leaf size=22 \[ -\frac {1}{2 b (\sinh (a+b x)+\cosh (a+b x))^2} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {4385} \[ -\frac {1}{2 b (\sinh (a+b x)+\cosh (a+b x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*b*(Cosh[a + b*x] + Sinh[a + b*x])^2)

Rule 4385

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[(q*A
ctivateTrig[y^(m + 1)])/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1] &&  !InertTrigFreeQ[u]

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 65, normalized size = 2.95 \[ -\frac {\sinh (2 a) \sinh (2 b x)}{2 b}-\frac {\cosh (2 a) \cosh (2 b x)}{2 b}+\frac {\sinh (2 a) \cosh (2 b x)}{2 b}+\frac {\cosh (2 a) \sinh (2 b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Cosh[2*a]*Cosh[2*b*x])/b + (Cosh[2*b*x]*Sinh[2*a])/(2*b) + (Cosh[2*a]*Sinh[2*b*x])/(2*b) - (Sinh[2*a]*Si
nh[2*b*x])/(2*b)

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fricas [A]  time = 0.41, size = 40, normalized size = 1.82 \[ -\frac {1}{2 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 14, normalized size = 0.64 \[ -\frac {e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-2*b*x - 2*a)/b

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maple [A]  time = 0.02, size = 36, normalized size = 1.64 \[ -\frac {\cosh \left (b x +a \right )-\sinh \left (b x +a \right )}{2 b \left (\cosh \left (b x +a \right )+\sinh \left (b x +a \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 14, normalized size = 0.64 \[ -\frac {e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-2*b*x - 2*a)/b

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mupad [B]  time = 0.10, size = 14, normalized size = 0.64 \[ -\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(- 2*a - 2*b*x)/(2*b)

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sympy [A]  time = 0.41, size = 37, normalized size = 1.68 \[ \begin {cases} \frac {\sinh {\left (a + b x \right )}}{b \sinh {\left (a + b x \right )} + b \cosh {\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \left (- \sinh {\relax (a )} + \cosh {\relax (a )}\right )}{\sinh {\relax (a )} + \cosh {\relax (a )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sinh(a + b*x)/(b*sinh(a + b*x) + b*cosh(a + b*x)), Ne(b, 0)), (x*(-sinh(a) + cosh(a))/(sinh(a) + co
sh(a)), True))

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