### 3.598 $$\int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx$$

Optimal. Leaf size=26 $\frac{(a \sinh (c+d x)+a \cosh (c+d x))^3}{3 d}$

[Out]

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

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Rubi [A]  time = 0.0151791, 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} $\frac{(a \sinh (c+d x)+a \cosh (c+d x))^3}{3 d}$

Antiderivative was successfully veriﬁed.

[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 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))^3 \, dx &=\frac{(a \cosh (c+d x)+a \sinh (c+d x))^3}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0763161, size = 25, normalized size = 0.96 $\frac{a^3 (\sinh (c+d x)+\cosh (c+d x))^3}{3 d}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{{a}^{3} \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.06453, size = 197, normalized size = 7.58 \begin{align*} \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 )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.95203, size = 163, normalized size = 6.27 \begin{align*} \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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [A]  time = 0.877147, size = 83, normalized size = 3.19 \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.15384, size = 23, normalized size = 0.88 \begin{align*} \frac{a^{3} e^{\left (3 \, d x + 3 \, c\right )}}{3 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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