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

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Rubi [A]  time = 0.0151521, 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} $\frac{(a \sinh (c+d x)+a \cosh (c+d x))^n}{d n}$

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*}

Mathematica [A]  time = 0.0741302, 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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Maple [A]  time = 0.009, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a\cosh \left ( dx+c \right ) +a\sinh \left ( dx+c \right ) \right ) ^{n}}{dn}} \end{align*}

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 = 1.02406, size = 24, normalized size = 0.92 \begin{align*} \frac{a^{n} e^{\left ({\left (d x + c\right )} n\right )}}{d n} \end{align*}

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 = 2.0289, size = 93, normalized size = 3.58 \begin{align*} \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{align*}

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 = 0.283659, size = 36, normalized size = 1.38 \begin{align*} \begin{cases} x & \text{for}\: d = 0 \wedge n = 0 \\x \left (a \sinh{\left (c \right )} + a \cosh{\left (c \right )}\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} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )}^{n}\,{d x} \end{align*}

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]

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