3.609 \(\int \frac {1}{a \cosh (c+d x)-a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \]

[Out]

1/d/(a*cosh(d*x+c)-a*sinh(d*x+c))

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3071} \[ \frac {1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[c + d*x] - a*Sinh[c + d*x])^(-1),x]

[Out]

1/(d*(a*Cosh[c + d*x] - a*Sinh[c + d*x]))

Rule 3071

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.92 \[ \frac {1}{a d \cosh (c+d x)-a d \sinh (c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[c + d*x] - a*Sinh[c + d*x])^(-1),x]

[Out]

(a*d*Cosh[c + d*x] - a*d*Sinh[c + d*x])^(-1)

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fricas [A]  time = 0.41, size = 20, normalized size = 0.83 \[ \frac {\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(d*x+c)-a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(cosh(d*x + c) + sinh(d*x + c))/(a*d)

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giac [A]  time = 0.14, size = 13, normalized size = 0.54 \[ \frac {e^{\left (d x + c\right )}}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(d*x+c)-a*sinh(d*x+c)),x, algorithm="giac")

[Out]

e^(d*x + c)/(a*d)

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maple [A]  time = 0.02, size = 25, normalized size = 1.04 \[ \frac {1}{d a \left (\cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(d*x+c)-a*sinh(d*x+c)),x)

[Out]

1/d/a/(cosh(d*x+c)-sinh(d*x+c))

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maxima [A]  time = 0.51, size = 13, normalized size = 0.54 \[ \frac {e^{\left (d x + c\right )}}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(d*x+c)-a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

e^(d*x + c)/(a*d)

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mupad [B]  time = 1.53, size = 13, normalized size = 0.54 \[ \frac {{\mathrm {e}}^{c+d\,x}}{a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(c + d*x) - a*sinh(c + d*x)),x)

[Out]

exp(c + d*x)/(a*d)

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sympy [A]  time = 0.35, size = 32, normalized size = 1.33 \[ \begin {cases} \frac {1}{- a d \sinh {\left (c + d x \right )} + a d \cosh {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x}{- a \sinh {\relax (c )} + a \cosh {\relax (c )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(d*x+c)-a*sinh(d*x+c)),x)

[Out]

Piecewise((1/(-a*d*sinh(c + d*x) + a*d*cosh(c + d*x)), Ne(d, 0)), (x/(-a*sinh(c) + a*cosh(c)), True))

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