3.282 \(\int e^x \cosh (4 x) \, dx\)

Optimal. Leaf size=19 \[ \frac {e^{5 x}}{10}-\frac {1}{6} e^{-3 x} \]

[Out]

-1/6/exp(3*x)+1/10*exp(5*x)

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2282, 12, 14} \[ \frac {e^{5 x}}{10}-\frac {1}{6} e^{-3 x} \]

Antiderivative was successfully verified.

[In]

Int[E^x*Cosh[4*x],x]

[Out]

-1/(6*E^(3*x)) + E^(5*x)/10

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {align*} \int e^x \cosh (4 x) \, dx &=\operatorname {Subst}\left (\int \frac {1+x^8}{2 x^4} \, dx,x,e^x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^8}{x^4} \, dx,x,e^x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^4}+x^4\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{6} e^{-3 x}+\frac {e^{5 x}}{10}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {e^{5 x}}{10}-\frac {1}{6} e^{-3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cosh[4*x],x]

[Out]

-1/6*1/E^(3*x) + E^(5*x)/10

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fricas [B]  time = 1.41, size = 46, normalized size = 2.42 \[ -\frac {\cosh \relax (x)^{4} - 16 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} - 16 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4}}{15 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x),x, algorithm="fricas")

[Out]

-1/15*(cosh(x)^4 - 16*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 16*cosh(x)*sinh(x)^3 + sinh(x)^4)/(cosh(x) -
sinh(x))

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giac [A]  time = 0.14, size = 13, normalized size = 0.68 \[ \frac {1}{10} \, e^{\left (5 \, x\right )} - \frac {1}{6} \, e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x),x, algorithm="giac")

[Out]

1/10*e^(5*x) - 1/6*e^(-3*x)

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maple [A]  time = 0.09, size = 26, normalized size = 1.37 \[ \frac {\sinh \left (3 x \right )}{6}+\frac {\sinh \left (5 x \right )}{10}-\frac {\cosh \left (3 x \right )}{6}+\frac {\cosh \left (5 x \right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cosh(4*x),x)

[Out]

1/6*sinh(3*x)+1/10*sinh(5*x)-1/6*cosh(3*x)+1/10*cosh(5*x)

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maxima [A]  time = 0.32, size = 13, normalized size = 0.68 \[ \frac {1}{10} \, e^{\left (5 \, x\right )} - \frac {1}{6} \, e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x),x, algorithm="maxima")

[Out]

1/10*e^(5*x) - 1/6*e^(-3*x)

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mupad [B]  time = 0.05, size = 14, normalized size = 0.74 \[ \frac {{\mathrm {e}}^{-3\,x}\,\left (3\,{\mathrm {e}}^{8\,x}-5\right )}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(4*x)*exp(x),x)

[Out]

(exp(-3*x)*(3*exp(8*x) - 5))/30

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sympy [A]  time = 0.25, size = 20, normalized size = 1.05 \[ \frac {4 e^{x} \sinh {\left (4 x \right )}}{15} - \frac {e^{x} \cosh {\left (4 x \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x),x)

[Out]

4*exp(x)*sinh(4*x)/15 - exp(x)*cosh(4*x)/15

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