\(\int e^x \sinh (4 x) \, dx\) [319]

Optimal result

Integrand size = 8, antiderivative size = 19 \[ \int e^x \sinh (4 x) \, dx=\frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \]

[Out]

1/6/exp(3*x)+1/10*exp(5*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} \[ \int e^x \sinh (4 x) \, dx=\frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \]

[In]

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

[Out]

1/(6*E^(3*x)) + E^(5*x)/10

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^8}{2 x^4} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-1+x^8}{x^4} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x^4}+x^4\right ) \, dx,x,e^x\right ) \\ & = \frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int e^x \sinh (4 x) \, dx=\frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \]

[In]

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

[Out]

1/(6*E^(3*x)) + E^(5*x)/10

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {{\mathrm e}^{5 x}}{10}+\frac {{\mathrm e}^{-3 x}}{6}\)	\(14\)
parallelrisch	\(\frac {{\mathrm e}^{x} \left (4 \cosh \left (4 x \right )-\sinh \left (4 x \right )\right )}{15}\)	\(18\)
default	\(-\frac {\sinh \left (3 x \right )}{6}+\frac {\sinh \left (5 x \right )}{10}+\frac {\cosh \left (3 x \right )}{6}+\frac {\cosh \left (5 x \right )}{10}\)	\(26\)

[In]

int(exp(x)*sinh(4*x),x,method=_RETURNVERBOSE)

[Out]

1/10*exp(5*x)+1/6*exp(-3*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42 \[ \int e^x \sinh (4 x) \, dx=\frac {4 \, {\left (\cosh \left (x\right )^{4} - \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}}{15 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]

[In]

integrate(exp(x)*sinh(4*x),x, algorithm="fricas")

[Out]

4/15*(cosh(x)^4 - cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - cosh(x)*sinh(x)^3 + sinh(x)^4)/(cosh(x) - sinh(x
))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^x \sinh (4 x) \, dx=- \frac {e^{x} \sinh {\left (4 x \right )}}{15} + \frac {4 e^{x} \cosh {\left (4 x \right )}}{15} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (4 x) \, dx=\frac {1}{10} \, e^{\left (5 \, x\right )} + \frac {1}{6} \, e^{\left (-3 \, x\right )} \]

[In]

integrate(exp(x)*sinh(4*x),x, algorithm="maxima")

[Out]

1/10*e^(5*x) + 1/6*e^(-3*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (4 x) \, dx=\frac {1}{10} \, e^{\left (5 \, x\right )} + \frac {1}{6} \, e^{\left (-3 \, x\right )} \]

[In]

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

[Out]

1/10*e^(5*x) + 1/6*e^(-3*x)

Mupad [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (4 x) \, dx=\frac {{\mathrm {e}}^{-3\,x}}{6}+\frac {{\mathrm {e}}^{5\,x}}{10} \]

[In]

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

[Out]

exp(-3*x)/6 + exp(5*x)/10