3.3.81 \(\int e^x \cosh ^2(4 x) \, dx\) [281]

Optimal. Leaf size=26 \[ -\frac {1}{28} e^{-7 x}+\frac {e^x}{2}+\frac {e^{9 x}}{36} \]

[Out]

-1/28/exp(7*x)+1/2*exp(x)+1/36*exp(9*x)

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Rubi [A]
time = 0.01, 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 = {2320, 12, 276} \begin {gather*} -\frac {1}{28} e^{-7 x}+\frac {e^x}{2}+\frac {e^{9 x}}{36} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*Cosh[4*x]^2,x]

[Out]

-1/28*1/E^(7*x) + E^x/2 + E^(9*x)/36

Rule 12

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

Rule 276

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 2320

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(4 x) \, dx &=\text {Subst}\left (\int \frac {\left (1+x^8\right )^2}{4 x^8} \, dx,x,e^x\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {\left (1+x^8\right )^2}{x^8} \, dx,x,e^x\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (2+\frac {1}{x^8}+x^8\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{28} e^{-7 x}+\frac {e^x}{2}+\frac {e^{9 x}}{36}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{28} e^{-7 x}+\frac {e^x}{2}+\frac {e^{9 x}}{36} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cosh[4*x]^2,x]

[Out]

-1/28*1/E^(7*x) + E^x/2 + E^(9*x)/36

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Maple [A]
time = 0.60, size = 34, normalized size = 1.31

method	result	size
risch	\(\frac {{\mathrm e}^{9 x}}{36}+\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-7 x}}{28}\)	\(18\)
default	\(\frac {\sinh \left (x \right )}{2}+\frac {\sinh \left (7 x \right )}{28}+\frac {\sinh \left (9 x \right )}{36}+\frac {\cosh \left (x \right )}{2}-\frac {\cosh \left (7 x \right )}{28}+\frac {\cosh \left (9 x \right )}{36}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cosh(4*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*sinh(x)+1/28*sinh(7*x)+1/36*sinh(9*x)+1/2*cosh(x)-1/28*cosh(7*x)+1/36*cosh(9*x)

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Maxima [A]
time = 0.27, size = 17, normalized size = 0.65 \begin {gather*} \frac {1}{36} \, e^{\left (9 \, x\right )} - \frac {1}{28} \, e^{\left (-7 \, x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x)^2,x, algorithm="maxima")

[Out]

1/36*e^(9*x) - 1/28*e^(-7*x) + 1/2*e^x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (17) = 34\).
time = 0.47, size = 87, normalized size = 3.35 \begin {gather*} -\frac {\cosh \left (x\right )^{8} - 64 \, \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} - 448 \, \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} - 448 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} - 64 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} - 63}{126 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x)^2,x, algorithm="fricas")

[Out]

-1/126*(cosh(x)^8 - 64*cosh(x)^7*sinh(x) + 28*cosh(x)^6*sinh(x)^2 - 448*cosh(x)^5*sinh(x)^3 + 70*cosh(x)^4*sin
h(x)^4 - 448*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 - 64*cosh(x)*sinh(x)^7 + sinh(x)^8 - 63)/(cosh(x) -
sinh(x))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
time = 0.17, size = 42, normalized size = 1.62 \begin {gather*} - \frac {32 e^{x} \sinh ^{2}{\left (4 x \right )}}{63} + \frac {8 e^{x} \sinh {\left (4 x \right )} \cosh {\left (4 x \right )}}{63} + \frac {31 e^{x} \cosh ^{2}{\left (4 x \right )}}{63} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x)**2,x)

[Out]

-32*exp(x)*sinh(4*x)**2/63 + 8*exp(x)*sinh(4*x)*cosh(4*x)/63 + 31*exp(x)*cosh(4*x)**2/63

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Giac [A]
time = 0.40, size = 17, normalized size = 0.65 \begin {gather*} \frac {1}{36} \, e^{\left (9 \, x\right )} - \frac {1}{28} \, e^{\left (-7 \, x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cosh(4*x)^2,x, algorithm="giac")

[Out]

1/36*e^(9*x) - 1/28*e^(-7*x) + 1/2*e^x

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Mupad [B]
time = 0.96, size = 17, normalized size = 0.65 \begin {gather*} \frac {{\mathrm {e}}^{9\,x}}{36}-\frac {{\mathrm {e}}^{-7\,x}}{28}+\frac {{\mathrm {e}}^x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(4*x)^2*exp(x),x)

[Out]

exp(9*x)/36 - exp(-7*x)/28 + exp(x)/2

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