3.608 \(\int (a \cosh (c+d x)-a \sinh (c+d x))^n \, dx\)

Optimal. Leaf size=28 \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^n}{d n} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3071} \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^n}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 27, normalized size = 0.96 \[ -\frac {(a (\cosh (c+d x)-\sinh (c+d x)))^n}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 39, normalized size = 1.39 \[ -\frac {\cosh \left (-d n x - c n + n \log \relax (a)\right ) + \sinh \left (-d n x - c n + n \log \relax (a)\right )}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(d*x+c)-a*sinh(d*x+c))^n,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 23, normalized size = 0.82 \[ -\frac {e^{\left (-d n x - c n + n \log \relax (a)\right )}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(d*x+c)-a*sinh(d*x+c))^n,x, algorithm="giac")

[Out]

-e^(-d*n*x - c*n + n*log(a))/(d*n)

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maple [A]  time = 0.02, size = 29, normalized size = 1.04 \[ -\frac {\left (a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )\right )^{n}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cosh(d*x+c)-a*sinh(d*x+c))^n,x)

[Out]

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

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maxima [A]  time = 0.49, size = 20, normalized size = 0.71 \[ -\frac {a^{n} e^{\left (-{\left (d x + c\right )} n\right )}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(d*x+c)-a*sinh(d*x+c))^n,x, algorithm="maxima")

[Out]

-a^n*e^(-(d*x + c)*n)/(d*n)

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mupad [B]  time = 1.66, size = 21, normalized size = 0.75 \[ -\frac {{\left (a\,{\mathrm {e}}^{-c-d\,x}\right )}^n}{d\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cosh(c + d*x) - a*sinh(c + d*x))^n,x)

[Out]

-(a*exp(- c - d*x))^n/(d*n)

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sympy [A]  time = 0.19, size = 37, normalized size = 1.32 \[ \begin {cases} x & \text {for}\: d = 0 \wedge n = 0 \\x \left (- a \sinh {\relax (c )} + a \cosh {\relax (c )}\right )^{n} & \text {for}\: d = 0 \\x & \text {for}\: n = 0 \\- \frac {\left (- a \sinh {\left (c + d x \right )} + a \cosh {\left (c + d x \right )}\right )^{n}}{d n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(d*x+c)-a*sinh(d*x+c))**n,x)

[Out]

Piecewise((x, Eq(d, 0) & Eq(n, 0)), (x*(-a*sinh(c) + a*cosh(c))**n, Eq(d, 0)), (x, Eq(n, 0)), (-(-a*sinh(c + d
*x) + a*cosh(c + d*x))**n/(d*n), True))

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