3.602 \(\int \frac{1}{(a \cosh (c+d x)+a \sinh (c+d x))^3} \, dx\)

Optimal. Leaf size=26 \[ -\frac{1}{3 d (a \sinh (c+d x)+a \cosh (c+d x))^3} \]

[Out]

-1/(3*d*(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3)

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Rubi [A]  time = 0.0158623, antiderivative size = 26, 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} \[ -\frac{1}{3 d (a \sinh (c+d x)+a \cosh (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[c + d*x] + a*Sinh[c + d*x])^(-3),x]

[Out]

-1/(3*d*(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3)

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))^3} \, dx &=-\frac{1}{3 d (a \cosh (c+d x)+a \sinh (c+d x))^3}\\ \end{align*}

Mathematica [A]  time = 0.0456126, size = 26, normalized size = 1. \[ -\frac{1}{3 d (a \sinh (c+d x)+a \cosh (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[c + d*x] + a*Sinh[c + d*x])^(-3),x]

[Out]

-1/(3*d*(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3)

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Maple [A]  time = 0., size = 24, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,d{a}^{3} \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) ^{3}}} \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))^3,x)

[Out]

-1/3/d/a^3/(cosh(d*x+c)+sinh(d*x+c))^3

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Maxima [A]  time = 1.07047, size = 23, normalized size = 0.88 \begin{align*} -\frac{e^{\left (-3 \, d x - 3 \, c\right )}}{3 \, a^{3} 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))^3,x, algorithm="maxima")

[Out]

-1/3*e^(-3*d*x - 3*c)/(a^3*d)

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Fricas [B]  time = 1.94219, size = 181, normalized size = 6.96 \begin{align*} -\frac{1}{3 \,{\left (a^{3} d \cosh \left (d x + c\right )^{3} + 3 \, a^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{3} d \sinh \left (d x + c\right )^{3}\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))^3,x, algorithm="fricas")

[Out]

-1/3/(a^3*d*cosh(d*x + c)^3 + 3*a^3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^3*d*cosh(d*x + c)*sinh(d*x + c)^2 +
a^3*d*sinh(d*x + c)^3)

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Sympy [A]  time = 2.20831, size = 90, normalized size = 3.46 \begin{align*} \begin{cases} - \frac{1}{3 a^{3} d \sinh ^{3}{\left (c + d x \right )} + 9 a^{3} d \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )} + 9 a^{3} d \sinh{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )} + 3 a^{3} d \cosh ^{3}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\left (a \sinh{\left (c \right )} + a \cosh{\left (c \right )}\right )^{3}} & \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))**3,x)

[Out]

Piecewise((-1/(3*a**3*d*sinh(c + d*x)**3 + 9*a**3*d*sinh(c + d*x)**2*cosh(c + d*x) + 9*a**3*d*sinh(c + d*x)*co
sh(c + d*x)**2 + 3*a**3*d*cosh(c + d*x)**3), Ne(d, 0)), (x/(a*sinh(c) + a*cosh(c))**3, True))

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Giac [A]  time = 1.17145, size = 23, normalized size = 0.88 \begin{align*} -\frac{e^{\left (-3 \, d x - 3 \, c\right )}}{3 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

-1/3*e^(-3*d*x - 3*c)/(a^3*d)