3.4.14 \(\int e^x \sinh ^2(3 x) \, dx\) [314]

Optimal. Leaf size=26 \[ -\frac {1}{20} e^{-5 x}-\frac {e^x}{2}+\frac {e^{7 x}}{28} \]

[Out]

-1/20/exp(5*x)-1/2*exp(x)+1/28*exp(7*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}{20} e^{-5 x}-\frac {e^x}{2}+\frac {e^{7 x}}{28} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*Sinh[3*x]^2,x]

[Out]

-1/20*1/E^(5*x) - E^x/2 + E^(7*x)/28

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 \sinh ^2(3 x) \, dx &=\text {Subst}\left (\int \frac {\left (1-x^6\right )^2}{4 x^6} \, dx,x,e^x\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {\left (1-x^6\right )^2}{x^6} \, dx,x,e^x\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-2+\frac {1}{x^6}+x^6\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{20} e^{-5 x}-\frac {e^x}{2}+\frac {e^{7 x}}{28}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{20} e^{-5 x}-\frac {e^x}{2}+\frac {e^{7 x}}{28} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sinh[3*x]^2,x]

[Out]

-1/20*1/E^(5*x) - E^x/2 + E^(7*x)/28

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Maple [A]
time = 0.43, size = 34, normalized size = 1.31

method	result	size
risch	\(\frac {{\mathrm e}^{7 x}}{28}-\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-5 x}}{20}\)	\(18\)
default	\(-\frac {\sinh \left (x \right )}{2}+\frac {\sinh \left (5 x \right )}{20}+\frac {\sinh \left (7 x \right )}{28}-\frac {\cosh \left (x \right )}{2}-\frac {\cosh \left (5 x \right )}{20}+\frac {\cosh \left (7 x \right )}{28}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sinh(3*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*sinh(x)+1/20*sinh(5*x)+1/28*sinh(7*x)-1/2*cosh(x)-1/20*cosh(5*x)+1/28*cosh(7*x)

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Maxima [A]
time = 0.27, size = 17, normalized size = 0.65 \begin {gather*} \frac {1}{28} \, e^{\left (7 \, x\right )} - \frac {1}{20} \, e^{\left (-5 \, x\right )} - \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sinh(3*x)^2,x, algorithm="maxima")

[Out]

1/28*e^(7*x) - 1/20*e^(-5*x) - 1/2*e^x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (17) = 34\).
time = 0.51, size = 67, normalized size = 2.58 \begin {gather*} -\frac {\cosh \left (x\right )^{6} - 36 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} - 120 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} - 36 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 35}{70 \, {\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(3*x)^2,x, algorithm="fricas")

[Out]

-1/70*(cosh(x)^6 - 36*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 - 120*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh
(x)^4 - 36*cosh(x)*sinh(x)^5 + sinh(x)^6 + 35)/(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 = 0.19, size = 42, normalized size = 1.62 \begin {gather*} \frac {17 e^{x} \sinh ^{2}{\left (3 x \right )}}{35} + \frac {6 e^{x} \sinh {\left (3 x \right )} \cosh {\left (3 x \right )}}{35} - \frac {18 e^{x} \cosh ^{2}{\left (3 x \right )}}{35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sinh(3*x)**2,x)

[Out]

17*exp(x)*sinh(3*x)**2/35 + 6*exp(x)*sinh(3*x)*cosh(3*x)/35 - 18*exp(x)*cosh(3*x)**2/35

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Giac [A]
time = 0.42, size = 17, normalized size = 0.65 \begin {gather*} \frac {1}{28} \, e^{\left (7 \, x\right )} - \frac {1}{20} \, e^{\left (-5 \, x\right )} - \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sinh(3*x)^2,x, algorithm="giac")

[Out]

1/28*e^(7*x) - 1/20*e^(-5*x) - 1/2*e^x

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Mupad [B]
time = 0.58, size = 17, normalized size = 0.65 \begin {gather*} \frac {{\mathrm {e}}^{7\,x}}{28}-\frac {{\mathrm {e}}^{-5\,x}}{20}-\frac {{\mathrm {e}}^x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(3*x)^2*exp(x),x)

[Out]

exp(7*x)/28 - exp(-5*x)/20 - exp(x)/2

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