### 3.973 $$\int e^{n \sinh (a+b x)} \cosh (a+b x) \, dx$$

Optimal. Leaf size=17 $\frac{e^{n \sinh (a+b x)}}{b n}$

[Out]

E^(n*Sinh[a + b*x])/(b*n)

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Rubi [A]  time = 0.0135215, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {4336, 2194} $\frac{e^{n \sinh (a+b x)}}{b n}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(n*Sinh[a + b*x])*Cosh[a + b*x],x]

[Out]

E^(n*Sinh[a + b*x])/(b*n)

Rule 4336

Int[Cosh[(c_.)*((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(
b*c), Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*
(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{n \sinh (a+b x)} \cosh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{e^{n \sinh (a+b x)}}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0160259, size = 17, normalized size = 1. $\frac{e^{n \sinh (a+b x)}}{b n}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(n*Sinh[a + b*x])*Cosh[a + b*x],x]

[Out]

E^(n*Sinh[a + b*x])/(b*n)

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Maple [A]  time = 0.006, size = 17, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{n\sinh \left ( bx+a \right ) }}}{bn}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*sinh(b*x+a))*cosh(b*x+a),x)

[Out]

exp(n*sinh(b*x+a))/b/n

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Maxima [A]  time = 1.00828, size = 22, normalized size = 1.29 \begin{align*} \frac{e^{\left (n \sinh \left (b x + a\right )\right )}}{b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*cosh(b*x+a),x, algorithm="maxima")

[Out]

e^(n*sinh(b*x + a))/(b*n)

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Fricas [A]  time = 2.03225, size = 74, normalized size = 4.35 \begin{align*} \frac{\cosh \left (n \sinh \left (b x + a\right )\right ) + \sinh \left (n \sinh \left (b x + a\right )\right )}{b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*cosh(b*x+a),x, algorithm="fricas")

[Out]

(cosh(n*sinh(b*x + a)) + sinh(n*sinh(b*x + a)))/(b*n)

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Sympy [A]  time = 1.14793, size = 36, normalized size = 2.12 \begin{align*} \begin{cases} x \cosh{\left (a \right )} & \text{for}\: b = 0 \wedge n = 0 \\x e^{n \sinh{\left (a \right )}} \cosh{\left (a \right )} & \text{for}\: b = 0 \\\frac{\sinh{\left (a + b x \right )}}{b} & \text{for}\: n = 0 \\\frac{e^{n \sinh{\left (a + b x \right )}}}{b n} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*cosh(b*x+a),x)

[Out]

Piecewise((x*cosh(a), Eq(b, 0) & Eq(n, 0)), (x*exp(n*sinh(a))*cosh(a), Eq(b, 0)), (sinh(a + b*x)/b, Eq(n, 0)),
(exp(n*sinh(a + b*x))/(b*n), True))

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Giac [A]  time = 1.15056, size = 39, normalized size = 2.29 \begin{align*} \frac{e^{\left (\frac{1}{2} \, n{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}\right )}}{b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*cosh(b*x+a),x, algorithm="giac")

[Out]

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