### 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[c + d*x] + a*Sinh[c + d*x])))

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Rubi [A]  time = 0.0164745, antiderivative size = 24, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0382466, size = 24, normalized size = 1. $-\frac{1}{d (a \sinh (c+d x)+a \cosh (c+d x))}$

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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Maple [A]  time = 0.003, size = 24, normalized size = 1. \begin{align*} -{\frac{1}{da \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) }} \end{align*}

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 = 1.08815, size = 23, normalized size = 0.96 \begin{align*} -\frac{e^{\left (-d x - c\right )}}{a d} \end{align*}

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 = 1.9821, size = 59, normalized size = 2.46 \begin{align*} -\frac{1}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )} \end{align*}

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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Sympy [A]  time = 0.624938, size = 34, normalized size = 1.42 \begin{align*} \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{align*}

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 = 1.12213, size = 23, normalized size = 0.96 \begin{align*} -\frac{e^{\left (-d x - c\right )}}{a d} \end{align*}

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)