3.948 \(\int e^{c+d x} \cosh (a+b x) \, dx\)

Optimal. Leaf size=54 \[ \frac {b e^{c+d x} \sinh (a+b x)}{b^2-d^2}-\frac {d e^{c+d x} \cosh (a+b x)}{b^2-d^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5475} \[ \frac {b e^{c+d x} \sinh (a+b x)}{b^2-d^2}-\frac {d e^{c+d x} \cosh (a+b x)}{b^2-d^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Cosh[a + b*x],x]

[Out]

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

Rule 5475

Int[Cosh[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)
, x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 38, normalized size = 0.70 \[ \frac {e^{c+d x} (b \sinh (a+b x)-d \cosh (a+b x))}{(b-d) (b+d)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x)*Cosh[a + b*x],x]

[Out]

(E^(c + d*x)*(-(d*Cosh[a + b*x]) + b*Sinh[a + b*x]))/((b - d)*(b + d))

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fricas [A]  time = 0.48, size = 97, normalized size = 1.80 \[ -\frac {d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left (d \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.13, size = 40, normalized size = 0.74 \[ \frac {e^{\left (b x + d x + a + c\right )}}{2 \, {\left (b + d\right )}} - \frac {e^{\left (-b x + d x - a + c\right )}}{2 \, {\left (b - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.22, size = 78, normalized size = 1.44 \[ \frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}-\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{2 \left (b -d \right )}+\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sinh(a-c+(b-d)*x)/(b-d)+1/2*sinh(a+c+(b+d)*x)/(b+d)-1/2*cosh(a-c+(b-d)*x)/(b-d)+1/2*cosh(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(exp(d*x+c)*cosh(b*x+a),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 = 1.75, size = 54, normalized size = 1.00 \[ -\frac {{\mathrm {e}}^{c-a-b\,x+d\,x}\,\left (b+d-b\,{\mathrm {e}}^{2\,a+2\,b\,x}+d\,{\mathrm {e}}^{2\,a+2\,b\,x}\right )}{2\,\left (b^2-d^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*exp(c + d*x),x)

[Out]

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

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sympy [A]  time = 2.10, size = 184, normalized size = 3.41 \[ \begin {cases} x e^{c} \cosh {\relax (a )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{c} e^{d x} \sinh {\left (a - d x \right )}}{2} + \frac {x e^{c} e^{d x} \cosh {\left (a - d x \right )}}{2} - \frac {e^{c} e^{d x} \sinh {\left (a - d x \right )}}{2 d} & \text {for}\: b = - d \\- \frac {x e^{c} e^{d x} \sinh {\left (a + d x \right )}}{2} + \frac {x e^{c} e^{d x} \cosh {\left (a + d x \right )}}{2} + \frac {e^{c} e^{d x} \sinh {\left (a + d x \right )}}{d} - \frac {e^{c} e^{d x} \cosh {\left (a + d x \right )}}{2 d} & \text {for}\: b = d \\\frac {b e^{c} e^{d x} \sinh {\left (a + b x \right )}}{b^{2} - d^{2}} - \frac {d e^{c} e^{d x} \cosh {\left (a + b x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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