3.18.14 \(\int \frac {e^{\frac {21-4 x \log (5) \log (2+x^2)}{4 x}} (-42-21 x^2-8 x^3 \log (5))}{8 x^2+4 x^4} \, dx\)

Optimal. Leaf size=20 \[ e^{\frac {21}{4 x}-\log (5) \log \left (2+x^2\right )} \]

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Rubi [A]  time = 0.55, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1593, 6706} \begin {gather*} e^{\left .\frac {21}{4}\right /x} 5^{-\log \left (x^2+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((21 - 4*x*Log[5]*Log[2 + x^2])/(4*x))*(-42 - 21*x^2 - 8*x^3*Log[5]))/(8*x^2 + 4*x^4),x]

[Out]

E^(21/(4*x))/5^Log[2 + x^2]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {21-4 x \log (5) \log \left (2+x^2\right )}{4 x}} \left (-42-21 x^2-8 x^3 \log (5)\right )}{x^2 \left (8+4 x^2\right )} \, dx\\ &=5^{-\log \left (2+x^2\right )} e^{\left .\frac {21}{4}\right /x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 20, normalized size = 1.00 \begin {gather*} e^{\frac {21}{4 x}-\log (5) \log \left (2+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((21 - 4*x*Log[5]*Log[2 + x^2])/(4*x))*(-42 - 21*x^2 - 8*x^3*Log[5]))/(8*x^2 + 4*x^4),x]

[Out]

E^(21/(4*x) - Log[5]*Log[2 + x^2])

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fricas [A]  time = 0.56, size = 19, normalized size = 0.95 \begin {gather*} e^{\left (-\frac {4 \, x \log \relax (5) \log \left (x^{2} + 2\right ) - 21}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3*log(5)-21*x^2-42)*exp(1/4*(-4*x*log(5)*log(x^2+2)+21)/x)/(4*x^4+8*x^2),x, algorithm="fricas"
)

[Out]

e^(-1/4*(4*x*log(5)*log(x^2 + 2) - 21)/x)

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giac [A]  time = 0.20, size = 17, normalized size = 0.85 \begin {gather*} e^{\left (-\log \relax (5) \log \left (x^{2} + 2\right ) + \frac {21}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3*log(5)-21*x^2-42)*exp(1/4*(-4*x*log(5)*log(x^2+2)+21)/x)/(4*x^4+8*x^2),x, algorithm="giac")

[Out]

e^(-log(5)*log(x^2 + 2) + 21/4/x)

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maple [A]  time = 0.27, size = 18, normalized size = 0.90

method	result	size
risch	\(\left (x^{2}+2\right )^{-\ln \relax (5)} {\mathrm e}^{\frac {21}{4 x}}\)	\(18\)
gosper	\({\mathrm e}^{-\frac {4 x \ln \relax (5) \ln \left (x^{2}+2\right )-21}{4 x}}\)	\(20\)
norman	\({\mathrm e}^{\frac {-4 x \ln \relax (5) \ln \left (x^{2}+2\right )+21}{4 x}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^3*ln(5)-21*x^2-42)*exp(1/4*(-4*x*ln(5)*ln(x^2+2)+21)/x)/(4*x^4+8*x^2),x,method=_RETURNVERBOSE)

[Out]

(x^2+2)^(-ln(5))*exp(21/4/x)

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maxima [A]  time = 0.65, size = 17, normalized size = 0.85 \begin {gather*} e^{\left (-\log \relax (5) \log \left (x^{2} + 2\right ) + \frac {21}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3*log(5)-21*x^2-42)*exp(1/4*(-4*x*log(5)*log(x^2+2)+21)/x)/(4*x^4+8*x^2),x, algorithm="maxima"
)

[Out]

e^(-log(5)*log(x^2 + 2) + 21/4/x)

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mupad [B]  time = 1.43, size = 17, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {e}}^{\frac {21}{4\,x}}}{{\left (x^2+2\right )}^{\ln \relax (5)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(x*log(5)*log(x^2 + 2) - 21/4)/x)*(8*x^3*log(5) + 21*x^2 + 42))/(8*x^2 + 4*x^4),x)

[Out]

exp(21/(4*x))/(x^2 + 2)^log(5)

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sympy [A]  time = 0.57, size = 17, normalized size = 0.85 \begin {gather*} e^{\frac {- x \log {\relax (5 )} \log {\left (x^{2} + 2 \right )} + \frac {21}{4}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**3*ln(5)-21*x**2-42)*exp(1/4*(-4*x*ln(5)*ln(x**2+2)+21)/x)/(4*x**4+8*x**2),x)

[Out]

exp((-x*log(5)*log(x**2 + 2) + 21/4)/x)

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