\(\int e^x \sinh (3 x) \, dx\) [315]

Optimal result

Integrand size = 8, antiderivative size = 19 \[ \int e^x \sinh (3 x) \, dx=\frac {e^{-2 x}}{4}+\frac {e^{4 x}}{8} \]

[Out]

1/4/exp(2*x)+1/8*exp(4*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 12, 14} \[ \int e^x \sinh (3 x) \, dx=\frac {e^{-2 x}}{4}+\frac {e^{4 x}}{8} \]

[In]

Int[E^x*Sinh[3*x],x]

[Out]

1/(4*E^(2*x)) + E^(4*x)/8

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int e^x \sinh (3 x) \, dx=\frac {1}{8} e^{-2 x} \left (2+e^{6 x}\right ) \]

[In]

Integrate[E^x*Sinh[3*x],x]

[Out]

(2 + E^(6*x))/(8*E^(2*x))

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {{\mathrm e}^{4 x}}{8}+\frac {{\mathrm e}^{-2 x}}{4}\)	\(14\)
parallelrisch	\(-\frac {{\mathrm e}^{x} \left (-3 \cosh \left (3 x \right )+\sinh \left (3 x \right )\right )}{8}\)	\(16\)
default	\(-\frac {\sinh \left (2 x \right )}{4}+\frac {\sinh \left (4 x \right )}{8}+\frac {\cosh \left (2 x \right )}{4}+\frac {\cosh \left (4 x \right )}{8}\)	\(26\)

[In]

int(exp(x)*sinh(3*x),x,method=_RETURNVERBOSE)

[Out]

1/8*exp(4*x)+1/4*exp(-2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int e^x \sinh (3 x) \, dx=\frac {3 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 9 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \sinh \left (x\right )^{3}}{8 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]

[In]

integrate(exp(x)*sinh(3*x),x, algorithm="fricas")

[Out]

1/8*(3*cosh(x)^3 - 3*cosh(x)^2*sinh(x) + 9*cosh(x)*sinh(x)^2 - sinh(x)^3)/(cosh(x) - sinh(x))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^x \sinh (3 x) \, dx=- \frac {e^{x} \sinh {\left (3 x \right )}}{8} + \frac {3 e^{x} \cosh {\left (3 x \right )}}{8} \]

[In]

integrate(exp(x)*sinh(3*x),x)

[Out]

-exp(x)*sinh(3*x)/8 + 3*exp(x)*cosh(3*x)/8

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (3 x) \, dx=\frac {1}{8} \, e^{\left (4 \, x\right )} + \frac {1}{4} \, e^{\left (-2 \, x\right )} \]

[In]

integrate(exp(x)*sinh(3*x),x, algorithm="maxima")

[Out]

1/8*e^(4*x) + 1/4*e^(-2*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (3 x) \, dx=\frac {1}{8} \, e^{\left (4 \, x\right )} + \frac {1}{4} \, e^{\left (-2 \, x\right )} \]

[In]

integrate(exp(x)*sinh(3*x),x, algorithm="giac")

[Out]

1/8*e^(4*x) + 1/4*e^(-2*x)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int e^x \sinh (3 x) \, dx=\frac {{\mathrm {e}}^{-2\,x}\,\left ({\mathrm {e}}^{6\,x}+2\right )}{8} \]

[In]

int(sinh(3*x)*exp(x),x)

[Out]

(exp(-2*x)*(exp(6*x) + 2))/8