3.597 \(\int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx\)

Optimal. Leaf size=26 \[ \frac{(a \sinh (c+d x)+a \cosh (c+d x))^2}{2 d} \]

[Out]

(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2/(2*d)

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Rubi [A]  time = 0.0162527, 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} \[ \frac{(a \sinh (c+d x)+a \cosh (c+d x))^2}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2,x]

[Out]

(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2/(2*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))^2 \, dx &=\frac{(a \cosh (c+d x)+a \sinh (c+d x))^2}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0507779, size = 25, normalized size = 0.96 \[ \frac{a^2 (\sinh (c+d x)+\cosh (c+d x))^2}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2,x]

[Out]

(a^2*(Cosh[c + d*x] + Sinh[c + d*x])^2)/(2*d)

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{{a}^{2} \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) ^{2}}{2\,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))^2,x)

[Out]

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

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Maxima [B]  time = 1.06098, size = 119, normalized size = 4.58 \begin{align*} \frac{1}{8} \, a^{2}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{8} \, a^{2}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac{a^{2} \cosh \left (d x + c\right )^{2}}{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))^2,x, algorithm="maxima")

[Out]

1/8*a^2*(4*x + e^(2*d*x + 2*c)/d - e^(-2*d*x - 2*c)/d) - 1/8*a^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d
) + a^2*cosh(d*x + c)^2/d

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Fricas [A]  time = 1.98951, size = 109, normalized size = 4.19 \begin{align*} \frac{a^{2} \cosh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )}{2 \,{\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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.485086, size = 44, normalized size = 1.69 \begin{align*} \begin{cases} \frac{a^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} + \frac{a^{2} \cosh ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sinh{\left (c \right )} + a \cosh{\left (c \right )}\right )^{2} & \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))**2,x)

[Out]

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

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Giac [A]  time = 1.13193, size = 23, normalized size = 0.88 \begin{align*} \frac{a^{2} e^{\left (2 \, d x + 2 \, c\right )}}{2 \, 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))^2,x, algorithm="giac")

[Out]

1/2*a^2*e^(2*d*x + 2*c)/d