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

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

[Out]

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

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Rubi [A]  time = 0.0186246, 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 = {5474} \[ \frac{b e^{a+b x} \sinh (c+d x)}{b^2-d^2}-\frac{d e^{a+b x} \cosh (c+d x)}{b^2-d^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)*Sinh[c + d*x],x]

[Out]

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

Rule 5474

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Cosh[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^{a+b x} \sinh (c+d x) \, dx &=-\frac{d e^{a+b x} \cosh (c+d x)}{b^2-d^2}+\frac{b e^{a+b x} \sinh (c+d x)}{b^2-d^2}\\ \end{align*}

Mathematica [A]  time = 0.0694216, size = 38, normalized size = 0.7 \[ \frac{e^{a+b x} (b \sinh (c+d x)-d \cosh (c+d x))}{(b-d) (b+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, 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(b*x+a)*sinh(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)-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(b*x+a)*sinh(d*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51805, size = 176, normalized size = 3.26 \begin{align*} -\frac{d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) -{\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{b^{2} - d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.03425, size = 167, normalized size = 3.09 \begin{align*} \begin{cases} x e^{a} \sinh{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x e^{a} e^{- d x} \sinh{\left (c + d x \right )}}{2} + \frac{x e^{a} e^{- d x} \cosh{\left (c + d x \right )}}{2} + \frac{e^{a} e^{- d x} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = - d \\\frac{x e^{a} e^{d x} \sinh{\left (c + d x \right )}}{2} - \frac{x e^{a} e^{d x} \cosh{\left (c + d x \right )}}{2} + \frac{e^{a} e^{d x} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = d \\\frac{b e^{a} e^{b x} \sinh{\left (c + d x \right )}}{b^{2} - d^{2}} - \frac{d e^{a} e^{b x} \cosh{\left (c + d 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(b*x+a)*sinh(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.2214, 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(b*x+a)*sinh(d*x+c),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)