3.318 \(\int e^x \sinh ^2(4 x) \, dx\)

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

[Out]

-1/(28*E^(7*x)) - E^x/2 + E^(9*x)/36

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Rubi [A]  time = 0.0208464, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 12, 270} \[ -\frac{1}{28} e^{-7 x}-\frac{e^x}{2}+\frac{e^{9 x}}{36} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(28*E^(7*x)) - E^x/2 + E^(9*x)/36

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]]]

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]

Rubi steps

\begin{align*} \int e^x \sinh ^2(4 x) \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^8\right )^2}{4 x^8} \, dx,x,e^x\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (1-x^8\right )^2}{x^8} \, dx,x,e^x\right )\\ &=\frac{1}{4} \operatorname{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*}

Mathematica [A]  time = 0.0159615, size = 26, normalized size = 1. \[ -\frac{1}{28} e^{-7 x}-\frac{e^x}{2}+\frac{e^{9 x}}{36} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(28*E^(7*x)) - E^x/2 + E^(9*x)/36

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Maple [A]  time = 0.013, size = 34, normalized size = 1.3 \begin{align*} -{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.05269, size = 23, normalized size = 0.88 \begin{align*} \frac{1}{36} \, e^{\left (9 \, x\right )} - \frac{1}{28} \, e^{\left (-7 \, x\right )} - \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sinh(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]  time = 2.04191, size = 311, normalized size = 11.96 \begin{align*} -\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sinh(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]  time = 0.955436, size = 42, normalized size = 1.62 \begin{align*} \frac{31 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{32 e^{x} \cosh ^{2}{\left (4 x \right )}}{63} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10597, size = 23, normalized size = 0.88 \begin{align*} \frac{1}{36} \, e^{\left (9 \, x\right )} - \frac{1}{28} \, e^{\left (-7 \, x\right )} - \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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