3.7.9 \(\int \frac {1}{a \cosh (c+d x)-a \sinh (c+d x)} \, dx\) [609]

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 = 0.01, 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 = {3150} \begin {gather*} \frac {1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \end {gather*}

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 3150

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 \begin {gather*} \frac {1}{a d \cosh (c+d x)-a d \sinh (c+d x)} \end {gather*}

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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Maple [A]
time = 6.76, size = 22, normalized size = 0.92

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c}}{a d}\)	\(14\)
derivativedivides	\(-\frac {2}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(22\)
default	\(-\frac {2}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(22\)
gosper	\(-\frac {1}{a d \left (\sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/a/(tanh(1/2*d*x+1/2*c)-1)

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Maxima [A]
time = 0.27, size = 13, normalized size = 0.54 \begin {gather*} \frac {e^{\left (d x + c\right )}}{a d} \end {gather*}

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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Fricas [A]
time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} \frac {\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}{a d} \end {gather*}

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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Sympy [A]
time = 0.19, size = 32, normalized size = 1.33 \begin {gather*} \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 {\left (c \right )} + a \cosh {\left (c \right )}} & \text {otherwise} \end {cases} \end {gather*}

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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Giac [A]
time = 0.41, size = 13, normalized size = 0.54 \begin {gather*} \frac {e^{\left (d x + c\right )}}{a d} \end {gather*}

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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Mupad [B]
time = 1.53, size = 13, normalized size = 0.54 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}}{a\,d} \end {gather*}

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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