\(\int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx\) [599]

Optimal result

Integrand size = 19, antiderivative size = 26 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\frac {(a \cosh (c+d x)+a \sinh (c+d x))^n}{d n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3150} \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\frac {(a \sinh (c+d x)+a \cosh (c+d x))^n}{d n} \]

[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*} \text {integral}& = \frac {(a \cosh (c+d x)+a \sinh (c+d x))^n}{d n} \\ \end{align*}

Mathematica [A] (verified)

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

[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)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

method	result	size
gosper	\(\frac {\left (a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )\right )^{n}}{d n}\)	\(27\)
derivativedivides	\(\frac {\left (a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )\right )^{n}}{d n}\)	\(27\)
default	\(\frac {\left (a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )\right )^{n}}{d n}\)	\(27\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\frac {\cosh \left (d n x + c n + n \log \left (a\right )\right ) + \sinh \left (d n x + c n + n \log \left (a\right )\right )}{d n} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\begin {cases} x & \text {for}\: d = 0 \wedge n = 0 \\x \left (a \sinh {\left (c \right )} + a \cosh {\left (c \right )}\right )^{n} & \text {for}\: d = 0 \\x & \text {for}\: n = 0 \\\frac {\left (a \sinh {\left (c + d x \right )} + a \cosh {\left (c + d x \right )}\right )^{n}}{d n} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\frac {a^{n} e^{\left ({\left (d x + c\right )} n\right )}}{d n} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\frac {\left (a e^{\left (d x + c\right )}\right )^{n}}{d n} \]

[In]

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

[Out]

(a*e^(d*x + c))^n/(d*n)

Mupad [B] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^n \, dx=\frac {{\left (a\,{\mathrm {e}}^{c+d\,x}\right )}^n}{d\,n} \]

[In]

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

[Out]

(a*exp(c + d*x))^n/(d*n)