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

Optimal. Leaf size=43 \[ \frac {\sinh (a+x (b-d)-c)}{2 (b-d)}+\frac {\sinh (a+x (b+d)+c)}{2 (b+d)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5614, 2637} \[ \frac {\sinh (a+x (b-d)-c)}{2 (b-d)}+\frac {\sinh (a+x (b+d)+c)}{2 (b+d)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]*Cosh[c + d*x],x]

[Out]

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

Rule 2637

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

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]))

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 43, normalized size = 1.00 \[ \frac {\sinh (a+x (b-d)-c)}{2 (b-d)}+\frac {\sinh (a+x (b+d)+c)}{2 (b+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 71, normalized size = 1.65 \[ \frac {b \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - d \cosh \left (b x + a\right ) \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*cosh(d*x + c)*sinh(b*x + a) - d*cosh(b*x + a)*sinh(d*x + c))/((b^2 - d^2)*cosh(b*x + a)^2 - (b^2 - d^2)*sin
h(b*x + a)^2)

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giac [B]  time = 0.13, size = 85, normalized size = 1.98 \[ \frac {e^{\left (b x + d x + a + c\right )}}{4 \, {\left (b + d\right )}} + \frac {e^{\left (b x - d x + a - c\right )}}{4 \, {\left (b - d\right )}} - \frac {e^{\left (-b x + d x - a + c\right )}}{4 \, {\left (b - d\right )}} - \frac {e^{\left (-b x - d x - a - c\right )}}{4 \, {\left (b + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.21, size = 40, normalized size = 0.93 \[ \frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(-d/b>0)', see `assume?` for mo
re details)Is -d/b equal to -1?

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mupad [B]  time = 0.14, size = 42, normalized size = 0.98 \[ \frac {b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )-d\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{b^2-d^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*cosh(c + d*x),x)

[Out]

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

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sympy [A]  time = 1.47, size = 153, normalized size = 3.56 \[ \begin {cases} x \cosh {\relax (a )} \cosh {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh {\left (a - d x \right )} \sinh {\left (c + d x \right )}}{2} + \frac {x \cosh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (a - d x \right )}}{2 d} & \text {for}\: b = - d \\- \frac {x \sinh {\left (a + d x \right )} \sinh {\left (c + d x \right )}}{2} + \frac {x \cosh {\left (a + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (a + d x \right )}}{2 d} & \text {for}\: b = d \\\frac {b \sinh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{b^{2} - d^{2}} - \frac {d \sinh {\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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