3.133 \(\int \cosh (a+b x) \cosh (c+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac{\sinh (a+2 b x+c)}{4 b}+\frac{1}{2} x \cosh (a-c) \]

[Out]

(x*Cosh[a - c])/2 + Sinh[a + c + 2*b*x]/(4*b)

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Rubi [A]  time = 0.0188788, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5614, 2637} \[ \frac{\sinh (a+2 b x+c)}{4 b}+\frac{1}{2} x \cosh (a-c) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]*Cosh[c + b*x],x]

[Out]

(x*Cosh[a - c])/2 + Sinh[a + c + 2*b*x]/(4*b)

Rule 5614

Int[Cosh[v_]^(p_.)*Cosh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Cosh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cosh (a+b x) \cosh (c+b x) \, dx &=\int \left (\frac{1}{2} \cosh (a-c)+\frac{1}{2} \cosh (a+c+2 b x)\right ) \, dx\\ &=\frac{1}{2} x \cosh (a-c)+\frac{1}{2} \int \cosh (a+c+2 b x) \, dx\\ &=\frac{1}{2} x \cosh (a-c)+\frac{\sinh (a+c+2 b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0226661, size = 26, normalized size = 0.96 \[ \frac{\sinh (a+2 b x+c)+2 b x \cosh (a-c)}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]*Cosh[c + b*x],x]

[Out]

(2*b*x*Cosh[a - c] + Sinh[a + c + 2*b*x])/(4*b)

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Maple [A]  time = 0.009, size = 24, normalized size = 0.9 \begin{align*}{\frac{x\cosh \left ( a-c \right ) }{2}}+{\frac{\sinh \left ( 2\,bx+a+c \right ) }{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*cosh(b*x+c),x)

[Out]

1/2*x*cosh(a-c)+1/4*sinh(2*b*x+a+c)/b

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Maxima [B]  time = 1.13596, size = 78, normalized size = 2.89 \begin{align*} \frac{{\left (b x + a\right )}{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{4 \, b} + \frac{e^{\left (2 \, b x + a + c\right )}}{8 \, b} - \frac{e^{\left (-2 \, b x - a - c\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.82386, size = 231, normalized size = 8.56 \begin{align*} \frac{2 \, b x \cosh \left (-a + c\right ) + 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (b x + c\right ) - \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right ) - \sinh \left (b x + c\right )^{2} \sinh \left (-a + c\right )}{4 \,{\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b*x*cosh(-a + c) + 2*cosh(b*x + c)*cosh(-a + c)*sinh(b*x + c) - cosh(b*x + c)^2*sinh(-a + c) - sinh(b*x
+ c)^2*sinh(-a + c))/(b*cosh(-a + c)^2 - b*sinh(-a + c)^2)

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Sympy [A]  time = 1.09124, size = 58, normalized size = 2.15 \begin{align*} \begin{cases} - \frac{x \sinh{\left (a + b x \right )} \sinh{\left (b x + c \right )}}{2} + \frac{x \cosh{\left (a + b x \right )} \cosh{\left (b x + c \right )}}{2} + \frac{\sinh{\left (a + b x \right )} \cosh{\left (b x + c \right )}}{2 b} & \text{for}\: b \neq 0 \\x \cosh{\left (a \right )} \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*cosh(b*x+c),x)

[Out]

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

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Giac [B]  time = 1.13134, size = 93, normalized size = 3.44 \begin{align*} \frac{2 \, b x{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} -{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (2 \, b x + 2 \, c\right )} + 1\right )} e^{\left (-2 \, b x - a - c\right )} + e^{\left (2 \, b x + a + c\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*b*x*(e^(2*a) + e^(2*c))*e^(-a - c) - (e^(2*b*x + 2*a) + e^(2*b*x + 2*c) + 1)*e^(-2*b*x - a - c) + e^(2*
b*x + a + c))/b