### 3.604 $$\int \frac{1}{\sqrt{a \cosh (c+d x)+a \sinh (c+d x)}} \, dx$$

Optimal. Leaf size=26 $-\frac{2}{d \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}}$

[Out]

-2/(d*Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0170826, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.048, Rules used = {3071} $-\frac{2}{d \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]],x]

[Out]

-2/(d*Sqrt[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}{\sqrt{a \cosh (c+d x)+a \sinh (c+d x)}} \, dx &=-\frac{2}{d \sqrt{a \cosh (c+d x)+a \sinh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0344189, size = 24, normalized size = 0.92 $-\frac{2}{d \sqrt{a (\sinh (c+d x)+\cosh (c+d x))}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]],x]

[Out]

-2/(d*Sqrt[a*(Cosh[c + d*x] + Sinh[c + d*x])])

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 1. \begin{align*} -2\,{\frac{1}{d\sqrt{a\cosh \left ( dx+c \right ) +a\sinh \left ( dx+c \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))^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.07291, size = 23, normalized size = 0.88 \begin{align*} -\frac{2 \, e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{\sqrt{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))^(1/2),x, algorithm="maxima")

[Out]

-2*e^(-1/2*d*x - 1/2*c)/(sqrt(a)*d)

________________________________________________________________________________________

Fricas [A]  time = 1.95519, size = 113, normalized size = 4.35 \begin{align*} -\frac{2 \, \sqrt{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}}{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))^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(a*cosh(d*x + c) + a*sinh(d*x + c))/(a*d*cosh(d*x + c) + a*d*sinh(d*x + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sinh{\left (c + d x \right )} + a \cosh{\left (c + d x \right )}}}\, dx \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))**(1/2),x)

[Out]

Integral(1/sqrt(a*sinh(c + d*x) + a*cosh(c + d*x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.14864, size = 23, normalized size = 0.88 \begin{align*} -\frac{2 \, e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{\sqrt{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))^(1/2),x, algorithm="giac")

[Out]

-2*e^(-1/2*d*x - 1/2*c)/(sqrt(a)*d)