\(\int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx\) [598]

Optimal result

Integrand size = 19, antiderivative size = 26 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {(a \cosh (c+d x)+a \sinh (c+d x))^3}{3 d} \]

[Out]

1/3*(a*cosh(d*x+c)+a*sinh(d*x+c))^3/d

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} \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {(a \sinh (c+d x)+a \cosh (c+d x))^3}{3 d} \]

[In]

Int[(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3,x]

[Out]

(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3/(3*d)

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))^3}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {a^3 (\cosh (c+d x)+\sinh (c+d x))^3}{3 d} \]

[In]

Integrate[(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3,x]

[Out]

(a^3*(Cosh[c + d*x] + Sinh[c + d*x])^3)/(3*d)

Maple [A] (verified)

Time = 6.51 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {a^{3} {\mathrm e}^{3 d x +3 c}}{3 d}\)	\(18\)
gosper	\(\frac {a^{3} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{3}}{3 d}\)	\(24\)
derivativedivides	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+a^{3} \cosh \left (d x +c \right )^{3}+a^{3} \sinh \left (d x +c \right )^{3}+a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\)	\(74\)
default	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+a^{3} \cosh \left (d x +c \right )^{3}+a^{3} \sinh \left (d x +c \right )^{3}+a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\)	\(74\)
parts	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )}{d}+\frac {a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}+\frac {a^{3} \cosh \left (d x +c \right )^{3}}{d}+\frac {a^{3} \sinh \left (d x +c \right )^{3}}{d}\)	\(82\)

[In]

int((a*cosh(d*x+c)+a*sinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/3/d*a^3*exp(3*d*x+3*c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (24) = 48\).

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {a^{3} \cosh \left (d x + c\right )^{2} + 2 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{3} \sinh \left (d x + c\right )^{2}}{3 \, {\left (d \cosh \left (d x + c\right ) - d \sinh \left (d x + c\right )\right )}} \]

[In]

integrate((a*cosh(d*x+c)+a*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3*(a^3*cosh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sinh(d*x + c)^2)/(d*cosh(d*x + c) - d*sinh(
d*x + c))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (20) = 40\).

Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} + \frac {a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sinh {\left (c \right )} + a \cosh {\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \]

[In]

integrate((a*cosh(d*x+c)+a*sinh(d*x+c))**3,x)

[Out]

Piecewise((a**3*sinh(c + d*x)**3/(3*d) + a**3*sinh(c + d*x)**2*cosh(c + d*x)/d + a**3*sinh(c + d*x)*cosh(c + d
*x)**2/d + a**3*cosh(c + d*x)**3/(3*d), Ne(d, 0)), (x*(a*sinh(c) + a*cosh(c))**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (24) = 48\).

Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.62 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {a^{3} \cosh \left (d x + c\right )^{3}}{d} + \frac {a^{3} \sinh \left (d x + c\right )^{3}}{d} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

[In]

integrate((a*cosh(d*x+c)+a*sinh(d*x+c))^3,x, algorithm="maxima")

[Out]

a^3*cosh(d*x + c)^3/d + a^3*sinh(d*x + c)^3/d + 1/24*a^3*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(-d*x - c)
/d - e^(-3*d*x - 3*c)/d) + 1/24*a^3*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)
/d)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {a^{3} e^{\left (3 \, d x + 3 \, c\right )}}{3 \, d} \]

[In]

integrate((a*cosh(d*x+c)+a*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

1/3*a^3*e^(3*d*x + 3*c)/d

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{3\,d} \]

[In]

int((a*cosh(c + d*x) + a*sinh(c + d*x))^3,x)

[Out]

(a^3*exp(3*c + 3*d*x))/(3*d)