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

Optimal. Leaf size=22 \[ \frac {e^{n \sinh (c (a+b x))}}{b c n} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {4336, 2194} \[ \frac {e^{n \sinh (c (a+b x))}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(n*Sinh[c*(a + b*x)])/(b*c*n)

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 23, normalized size = 1.05 \[ \frac {e^{n \sinh (a c+b c x)}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(n*Sinh[a*c + b*c*x])/(b*c*n)

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fricas [A]  time = 0.48, size = 35, normalized size = 1.59 \[ \frac {\cosh \left (n \sinh \left (b c x + a c\right )\right ) + \sinh \left (n \sinh \left (b c x + a c\right )\right )}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 38, normalized size = 1.73 \[ \frac {e^{\left (\frac {1}{2} \, n e^{\left (b c x + a c\right )} - \frac {1}{2} \, n e^{\left (-b c x - a c\right )}\right )}}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.84, size = 39, normalized size = 1.77 \[ \frac {\frac {\sinh \left (n \sinh \left (c \left (b x +a \right )\right )\right )}{n}+\frac {\cosh \left (n \sinh \left (c \left (b x +a \right )\right )\right )}{n}}{c b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c/b*(1/n*sinh(n*sinh(c*(b*x+a)))+cosh(n*sinh(c*(b*x+a)))/n)

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maxima [A]  time = 0.30, size = 22, normalized size = 1.00 \[ \frac {e^{\left (n \sinh \left (b c x + a c\right )\right )}}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(n*sinh(b*c*x + a*c))/(b*c*n)

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mupad [B]  time = 1.71, size = 38, normalized size = 1.73 \[ \frac {{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}}\,{\mathrm {e}}^{-\frac {n\,{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}}{b\,c\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp((n*exp(b*c*x)*exp(a*c))/2)*exp(-(n*exp(-b*c*x)*exp(-a*c))/2))/(b*c*n)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \sinh {\left (a c + b c x \right )}} \cosh {\left (a c + b c x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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