\(\int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx\) [596]

Optimal result

Integrand size = 17, antiderivative size = 23 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh (c+d x)}{d}+\frac {a \sinh (c+d x)}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2717, 2718} \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \sinh (c+d x)}{d}+\frac {a \cosh (c+d x)}{d} \]

[In]

Int[a*Cosh[c + d*x] + a*Sinh[c + d*x],x]

[Out]

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = a \int \cosh (c+d x) \, dx+a \int \sinh (c+d x) \, dx \\ & = \frac {a \cosh (c+d x)}{d}+\frac {a \sinh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh (c) \cosh (d x)}{d}+\frac {a \cosh (d x) \sinh (c)}{d}+\frac {a \cosh (c) \sinh (d x)}{d}+\frac {a \sinh (c) \sinh (d x)}{d} \]

[In]

Integrate[a*Cosh[c + d*x] + a*Sinh[c + d*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} a}{d}\)	\(12\)
gosper	\(\frac {a \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )}{d}\)	\(19\)
derivativedivides	\(\frac {a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )}{d}\)	\(22\)
default	\(\frac {a \cosh \left (d x +c \right )}{d}+\frac {a \sinh \left (d x +c \right )}{d}\)	\(24\)
parts	\(\frac {a \cosh \left (d x +c \right )}{d}+\frac {a \sinh \left (d x +c \right )}{d}\)	\(24\)
meijerg	\(\frac {\left (\cosh \left (c \right ) \sqrt {\pi }\, a +\sqrt {\pi }\, \sinh \left (c \right ) a \right ) \sinh \left (d x \right )}{d \sqrt {\pi }}-\frac {i \left (-i \cosh \left (c \right ) \sqrt {\pi }\, a -i \sqrt {\pi }\, \sinh \left (c \right ) a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}\)	\(66\)

[In]

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

[Out]

1/d*exp(d*x+c)*a

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}{d} \]

[In]

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

[Out]

(a*cosh(d*x + c) + a*sinh(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=a \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\left (c \right )} & \text {otherwise} \end {cases}\right ) + a \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\left (c \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a*Piecewise((sinh(c + d*x)/d, Ne(d, 0)), (x*cosh(c), True)) + a*Piecewise((cosh(c + d*x)/d, Ne(d, 0)), (x*sinh
(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh \left (d x + c\right )}{d} + \frac {a \sinh \left (d x + c\right )}{d} \]

[In]

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

[Out]

a*cosh(d*x + c)/d + a*sinh(d*x + c)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).

Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]

[In]

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

[Out]

1/2*a*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/2*a*(e^(d*x + c)/d - e^(-d*x - c)/d)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a\,{\mathrm {e}}^{c+d\,x}}{d} \]

[In]

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

[Out]

(a*exp(c + d*x))/d