### 3.969 $$\int e^{n \cosh (c (a+b x))} \sinh (a c+b c x) \, dx$$

Optimal. Leaf size=23 $\frac{e^{n \cosh (a c+b c x)}}{b c n}$

[Out]

E^(n*Cosh[a*c + b*c*x])/(b*c*n)

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Rubi [A]  time = 0.0151182, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {4337, 2194} $\frac{e^{n \cosh (a c+b c x)}}{b c n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

E^(n*Cosh[a*c + b*c*x])/(b*c*n)

Rule 4337

Int[(u_)*Sinh[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[d/(
b*c), Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[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 \cosh (c (a+b x))} \sinh (a c+b c x) \, dx &=\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\cosh (a c+b c x)\right )}{b c}\\ &=\frac{e^{n \cosh (a c+b c x)}}{b c n}\\ \end{align*}

Mathematica [A]  time = 0.0437569, size = 22, normalized size = 0.96 $\frac{e^{n \cosh (c (a+b x))}}{b c n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

E^(n*Cosh[c*(a + b*x)])/(b*c*n)

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Maple [A]  time = 0.009, size = 23, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{n\cosh \left ( bcx+ac \right ) }}}{cbn}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(n*cosh(b*c*x+a*c))/b/c/n

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Maxima [A]  time = 1.00429, size = 30, normalized size = 1.3 \begin{align*} \frac{e^{\left (n \cosh \left (b c x + a c\right )\right )}}{b c n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(n*cosh(b*c*x + a*c))/(b*c*n)

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Fricas [A]  time = 2.26688, size = 88, normalized size = 3.83 \begin{align*} \frac{\cosh \left (n \cosh \left (b c x + a c\right )\right ) + \sinh \left (n \cosh \left (b c x + a c\right )\right )}{b c n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.98863, size = 48, normalized size = 2.09 \begin{align*} \begin{cases} 0 & \text{for}\: b = 0 \wedge c = 0 \wedge n = 0 \\x e^{n \cosh{\left (a c \right )}} \sinh{\left (a c \right )} & \text{for}\: b = 0 \\\frac{\cosh{\left (a c + b c x \right )}}{b c} & \text{for}\: n = 0 \\0 & \text{for}\: c = 0 \\\frac{e^{n \cosh{\left (a c + b c x \right )}}}{b c n} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (n \cosh \left ({\left (b x + a\right )} c\right )\right )} \sinh \left (b c x + a c\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(n*cosh((b*x + a)*c))*sinh(b*c*x + a*c), x)