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

Optimal. Leaf size=27 \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^2}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3071} \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^2}{2 d} \]

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*}

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.00 \[ -\frac {(a \cosh (c+d x)-a \sinh (c+d x))^2}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*Cosh[c + d*x] - a*Sinh[c + d*x])^2/d

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fricas [A]  time = 0.38, size = 43, normalized size = 1.59 \[ -\frac {a^{2}}{2 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \]

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

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giac [A]  time = 0.14, size = 17, normalized size = 0.63 \[ -\frac {a^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, d} \]

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

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maple [A]  time = 0.02, size = 26, normalized size = 0.96 \[ -\frac {a^{2} \left (\cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )^{2}}{2 d} \]

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 = 0.30, size = 89, normalized size = 3.30 \[ \frac {1}{8} \, a^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {a^{2} \cosh \left (d x + c\right )^{2}}{d} \]

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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mupad [B]  time = 0.06, size = 17, normalized size = 0.63 \[ -\frac {a^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cosh(c + d*x) - a*sinh(c + d*x))^2,x)

[Out]

-(a^2*exp(- 2*c - 2*d*x))/(2*d)

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sympy [A]  time = 0.21, size = 44, normalized size = 1.63 \[ \begin {cases} - \frac {a^{2} \sinh ^{2}{\left (c + d x \right )}}{d} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (- a \sinh {\relax (c )} + a \cosh {\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]

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

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