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

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

[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 = 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} \[ \frac {(a \sinh (c+d x)+a \cosh (c+d x))^n}{d n} \]

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 = 0.08, size = 24, normalized size = 0.92 \[ \frac {(a (\sinh (c+d x)+\cosh (c+d x)))^n}{d n} \]

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 = 0.42, size = 34, normalized size = 1.31 \[ \frac {\cosh \left (d n x + c n + n \log \relax (a)\right ) + \sinh \left (d n x + c n + n \log \relax (a)\right )}{d n} \]

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 = 0.12, size = 20, normalized size = 0.77 \[ \frac {e^{\left (d n x + c n + n \log \relax (a)\right )}}{d n} \]

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 = 27, 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 = 0.57, size = 18, normalized size = 0.69 \[ \frac {a^{n} e^{\left ({\left (d x + c\right )} n\right )}}{d n} \]

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 = 1.63, size = 17, normalized size = 0.65 \[ \frac {{\left (a\,{\mathrm {e}}^{c+d\,x}\right )}^n}{d\,n} \]

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 = 36, normalized size = 1.38 \[ \begin {cases} x & \text {for}\: d = 0 \wedge n = 0 \\x \left (a \sinh {\relax (c )} + a \cosh {\relax (c )}\right )^{n} & \text {for}\: d = 0 \\x & \text {for}\: n = 0 \\\frac {\left (a \sinh {\left (c + d x \right )} + a \cosh {\left (c + d x \right )}\right )^{n}}{d n} & \text {otherwise} \end {cases} \]

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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