\(\int (a+b \sinh (c+d x)) \, dx\) [100]

Optimal result

Integrand size = 10, antiderivative size = 15 \[ \int (a+b \sinh (c+d x)) \, dx=a x+\frac {b \cosh (c+d x)}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2718} \[ \int (a+b \sinh (c+d x)) \, dx=a x+\frac {b \cosh (c+d x)}{d} \]

[In]

Int[a + b*Sinh[c + d*x],x]

[Out]

a*x + (b*Cosh[c + d*x])/d

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 x+b \int \sinh (c+d x) \, dx \\ & = a x+\frac {b \cosh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int (a+b \sinh (c+d x)) \, dx=a x+\frac {b \cosh (c) \cosh (d x)}{d}+\frac {b \sinh (c) \sinh (d x)}{d} \]

[In]

Integrate[a + b*Sinh[c + d*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
default	\(a x +\frac {b \cosh \left (d x +c \right )}{d}\)	\(16\)
parts	\(a x +\frac {b \cosh \left (d x +c \right )}{d}\)	\(16\)
parallelrisch	\(\frac {b \left (1+\cosh \left (d x +c \right )\right )}{d}+a x\)	\(18\)
derivativedivides	\(\frac {\left (d x +c \right ) a +b \cosh \left (d x +c \right )}{d}\)	\(21\)
risch	\(a x +\frac {b \,{\mathrm e}^{d x +c}}{2 d}+\frac {b \,{\mathrm e}^{-d x -c}}{2 d}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int (a+b \sinh (c+d x)) \, dx=\frac {a d x + b \cosh \left (d x + c\right )}{d} \]

[In]

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

[Out]

(a*d*x + b*cosh(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int (a+b \sinh (c+d x)) \, dx=a x + b \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\left (c \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a*x + b*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 = 15, normalized size of antiderivative = 1.00 \[ \int (a+b \sinh (c+d x)) \, dx=a x + \frac {b \cosh \left (d x + c\right )}{d} \]

[In]

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

[Out]

a*x + b*cosh(d*x + c)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int (a+b \sinh (c+d x)) \, dx=a x + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int (a+b \sinh (c+d x)) \, dx=a\,x+\frac {b\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]

[In]

int(a + b*sinh(c + d*x),x)

[Out]

a*x + (b*cosh(c + d*x))/d