3.273 \(\int e^x \cosh ^2(2 x) \, dx\)

Optimal. Leaf size=26 \[ -\frac {1}{12} e^{-3 x}+\frac {e^x}{2}+\frac {e^{5 x}}{20} \]

[Out]

-1/12/exp(3*x)+1/2*exp(x)+1/20*exp(5*x)

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2282, 12, 270} \[ -\frac {1}{12} e^{-3 x}+\frac {e^x}{2}+\frac {e^{5 x}}{20} \]

Antiderivative was successfully verified.

[In]

Int[E^x*Cosh[2*x]^2,x]

[Out]

-1/(12*E^(3*x)) + E^x/2 + E^(5*x)/20

Rule 12

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 ^2(2 x) \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^2}{4 x^4} \, dx,x,e^x\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^2}{x^4} \, dx,x,e^x\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (2+\frac {1}{x^4}+x^4\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{12} e^{-3 x}+\frac {e^x}{2}+\frac {e^{5 x}}{20}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \[ -\frac {1}{12} e^{-3 x}+\frac {e^x}{2}+\frac {e^{5 x}}{20} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cosh[2*x]^2,x]

[Out]

-1/12*1/E^(3*x) + E^x/2 + E^(5*x)/20

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fricas [B]  time = 0.44, size = 47, normalized size = 1.81 \[ -\frac {\cosh \relax (x)^{4} - 16 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} - 16 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 15}{30 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(2*x)^2,x, algorithm="fricas")

[Out]

-1/30*(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 - 15)/(cosh
(x) - sinh(x))

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giac [A]  time = 0.14, size = 17, normalized size = 0.65 \[ \frac {1}{20} \, e^{\left (5 \, x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{2} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(2*x)^2,x, algorithm="giac")

[Out]

1/20*e^(5*x) - 1/12*e^(-3*x) + 1/2*e^x

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maple [A]  time = 0.10, size = 34, normalized size = 1.31 \[ \frac {\sinh \relax (x )}{2}+\frac {\sinh \left (3 x \right )}{12}+\frac {\sinh \left (5 x \right )}{20}+\frac {\cosh \relax (x )}{2}-\frac {\cosh \left (3 x \right )}{12}+\frac {\cosh \left (5 x \right )}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cosh(2*x)^2,x)

[Out]

1/2*sinh(x)+1/12*sinh(3*x)+1/20*sinh(5*x)+1/2*cosh(x)-1/12*cosh(3*x)+1/20*cosh(5*x)

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maxima [A]  time = 0.31, size = 17, normalized size = 0.65 \[ \frac {1}{20} \, e^{\left (5 \, x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{2} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(2*x)^2,x, algorithm="maxima")

[Out]

1/20*e^(5*x) - 1/12*e^(-3*x) + 1/2*e^x

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mupad [B]  time = 0.08, size = 17, normalized size = 0.65 \[ \frac {{\mathrm {e}}^{5\,x}}{20}-\frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(2*x)^2*exp(x),x)

[Out]

exp(5*x)/20 - exp(-3*x)/12 + exp(x)/2

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sympy [B]  time = 0.65, size = 42, normalized size = 1.62 \[ - \frac {8 e^{x} \sinh ^{2}{\left (2 x \right )}}{15} + \frac {4 e^{x} \sinh {\left (2 x \right )} \cosh {\left (2 x \right )}}{15} + \frac {7 e^{x} \cosh ^{2}{\left (2 x \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(2*x)**2,x)

[Out]

-8*exp(x)*sinh(2*x)**2/15 + 4*exp(x)*sinh(2*x)*cosh(2*x)/15 + 7*exp(x)*cosh(2*x)**2/15

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