### 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*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)

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Rubi [A]  time = 0.0168503, antiderivative size = 54, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.030573, size = 38, normalized size = 0.7 $\frac{e^{c+d x} (b \sinh (a+b x)-d \cosh (a+b x))}{(b-d) (b+d)}$

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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Maple [A]  time = 0.005, size = 78, normalized size = 1.4 \begin{align*}{\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\,b-2\,d}}+{\frac{\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{2\,b+2\,d}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 1.84209, size = 240, normalized size = 4.44 \begin{align*} -\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}} \end{align*}

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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Sympy [A]  time = 4.72467, size = 201, normalized size = 3.72 \begin{align*} \begin{cases} x e^{c} \cosh{\left (a \right )} & \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 )}}{d} - \frac{e^{c} e^{d x} \cosh{\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} \end{align*}

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)/d - exp(c)*exp(d*x)*cosh(a - d*x)/(2*d), Eq(b, -d)), (-x*exp(c)*ex
p(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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Giac [A]  time = 1.27937, size = 54, normalized size = 1. \begin{align*} \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 )}} \end{align*}

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)