3.7.8 \(\int (a \cosh (c+d x)-a \sinh (c+d x))^n \, dx\) [608]

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 = 0.01, 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 = {3150} \begin {gather*} -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^n}{d n} \end {gather*}

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 3150

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 = 0.03, size = 27, normalized size = 0.96 \begin {gather*} -\frac {(a (\cosh (c+d x)-\sinh (c+d x)))^n}{d n} \end {gather*}

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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Maple [A]
time = 3.57, size = 29, normalized size = 1.04

method	result	size
gosper	\(-\frac {\left (a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )\right )^{n}}{d n}\)	\(29\)
derivativedivides	\(-\frac {\left (a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )\right )^{n}}{d n}\)	\(29\)
default	\(-\frac {\left (a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )\right )^{n}}{d n}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 20, normalized size = 0.71 \begin {gather*} -\frac {a^{n} e^{\left (-{\left (d x + c\right )} n\right )}}{d n} \end {gather*}

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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Fricas [A]
time = 0.38, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\cosh \left (-d n x - c n + n \log \left (a\right )\right ) + \sinh \left (-d n x - c n + n \log \left (a\right )\right )}{d n} \end {gather*}

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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Sympy [A]
time = 0.08, size = 39, normalized size = 1.39 \begin {gather*} \begin {cases} x & \text {for}\: n = 0 \wedge \left (d = 0 \vee n = 0\right ) \\x \left (- a \sinh {\left (c \right )} + a \cosh {\left (c \right )}\right )^{n} & \text {for}\: d = 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} \end {gather*}

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

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Giac [A]
time = 0.43, size = 23, normalized size = 0.82 \begin {gather*} -\frac {e^{\left (-d n x - c n + n \log \left (a\right )\right )}}{d n} \end {gather*}

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

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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