3.4.11 \(\int e^x \sinh (2 x) \, dx\) [311]

Optimal. Leaf size=19 \[ \frac {e^{-x}}{2}+\frac {e^{3 x}}{6} \]

[Out]

1/2/exp(x)+1/6*exp(3*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 = {2320, 12, 14} \begin {gather*} \frac {e^{-x}}{2}+\frac {e^{3 x}}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(2*E^x) + E^(3*x)/6

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 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 \sinh (2 x) \, dx &=\text {Subst}\left (\int \frac {-1+x^4}{2 x^2} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {-1+x^4}{x^2} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x^2}+x^2\right ) \, dx,x,e^x\right )\\ &=\frac {e^{-x}}{2}+\frac {e^{3 x}}{6}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{6} e^{-x} \left (3+e^{4 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3 + E^(4*x))/(6*E^x)

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Maple [A]
time = 0.25, size = 22, normalized size = 1.16

method	result	size
risch	\(\frac {{\mathrm e}^{3 x}}{6}+\frac {{\mathrm e}^{-x}}{2}\)	\(14\)
default	\(-\frac {\sinh \left (x \right )}{2}+\frac {\sinh \left (3 x \right )}{6}+\frac {\cosh \left (x \right )}{2}+\frac {\cosh \left (3 x \right )}{6}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sinh(2*x),x,method=_RETURNVERBOSE)

[Out]

-1/2*sinh(x)+1/6*sinh(3*x)+1/2*cosh(x)+1/6*cosh(3*x)

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{6} \, e^{\left (3 \, x\right )} + \frac {1}{2} \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*e^(3*x) + 1/2*e^(-x)

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Fricas [A]
time = 0.43, size = 26, normalized size = 1.37 \begin {gather*} \frac {2 \, {\left (\cosh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}}{3 \, {\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)*sinh(2*x),x, algorithm="fricas")

[Out]

2/3*(cosh(x)^2 - cosh(x)*sinh(x) + sinh(x)^2)/(cosh(x) - sinh(x))

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Sympy [A]
time = 0.10, size = 20, normalized size = 1.05 \begin {gather*} - \frac {e^{x} \sinh {\left (2 x \right )}}{3} + \frac {2 e^{x} \cosh {\left (2 x \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x)*sinh(2*x)/3 + 2*exp(x)*cosh(2*x)/3

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Giac [A]
time = 0.40, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{6} \, e^{\left (3 \, x\right )} + \frac {1}{2} \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*e^(3*x) + 1/2*e^(-x)

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Mupad [B]
time = 0.56, size = 12, normalized size = 0.63 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{4\,x}+3\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(-x)*(exp(4*x) + 3))/6

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