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

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

[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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3071} \[ -\frac {1}{d (a \sinh (c+d x)+a \cosh (c+d x))} \]

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 = 0.04, size = 24, normalized size = 1.00 \[ -\frac {1}{d (a \sinh (c+d x)+a \cosh (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(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])))

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fricas [A]  time = 0.40, size = 23, normalized size = 0.96 \[ -\frac {1}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )} \]

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]

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

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giac [A]  time = 0.13, size = 17, normalized size = 0.71 \[ -\frac {e^{\left (-d x - c\right )}}{a d} \]

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 = 24, normalized size = 1.00 \[ -\frac {1}{d a \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )} \]

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 = 0.31, size = 17, normalized size = 0.71 \[ -\frac {e^{\left (-d x - c\right )}}{a d} \]

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 = 0.07, size = 17, normalized size = 0.71 \[ -\frac {{\mathrm {e}}^{-c-d\,x}}{a\,d} \]

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 = 0.45, size = 34, normalized size = 1.42 \[ \begin {cases} - \frac {1}{a d \sinh {\left (c + d x \right )} + a d \cosh {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x}{a \sinh {\relax (c )} + a \cosh {\relax (c )}} & \text {otherwise} \end {cases} \]

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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