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(d*x+c)+a*sinh(d*x+c))^3

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Rubi [A]  time = 0.02, antiderivative size = 26, 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} \[ -\frac {1}{3 d (a \sinh (c+d x)+a \cosh (c+d x))^3} \]

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*}

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Mathematica [A]  time = 0.05, size = 26, normalized size = 1.00 \[ -\frac {1}{3 d (a \sinh (c+d x)+a \cosh (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 71, normalized size = 2.73 \[ -\frac {1}{3 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} + 3 \, a^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{3} d \sinh \left (d x + c\right )^{3}\right )}} \]

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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giac [A]  time = 0.13, size = 17, normalized size = 0.65 \[ -\frac {e^{\left (-3 \, d x - 3 \, c\right )}}{3 \, a^{3} d} \]

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)

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maple [A]  time = 0.02, size = 24, normalized size = 0.92 \[ -\frac {1}{3 d \,a^{3} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{3}} \]

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 = 0.32, size = 17, normalized size = 0.65 \[ -\frac {e^{\left (-3 \, d x - 3 \, c\right )}}{3 \, a^{3} d} \]

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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mupad [B]  time = 1.47, size = 17, normalized size = 0.65 \[ -\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{3\,a^3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(- 3*c - 3*d*x)/(3*a^3*d)

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sympy [A]  time = 1.42, size = 90, normalized size = 3.46 \[ \begin {cases} - \frac {1}{3 a^{3} d \sinh ^{3}{\left (c + d x \right )} + 9 a^{3} d \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )} + 9 a^{3} d \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )} + 3 a^{3} d \cosh ^{3}{\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \sinh {\relax (c )} + a \cosh {\relax (c )}\right )^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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