3.1.46 \(\int \frac {34-2 e^{4/3}+2 \log (2)}{1+4 x+4 x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {-17+e^{4/3}-\log (2)}{1+2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 27, 32} \begin {gather*} -\frac {17-e^{4/3}+\log (2)}{2 x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(34 - 2*E^(4/3) + 2*Log[2])/(1 + 4*x + 4*x^2),x]

[Out]

-((17 - E^(4/3) + Log[2])/(1 + 2*x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (34-2 e^{4/3}+2 \log (2)\right ) \int \frac {1}{1+4 x+4 x^2} \, dx\\ &=\left (34-2 e^{4/3}+2 \log (2)\right ) \int \frac {1}{(1+2 x)^2} \, dx\\ &=-\frac {17-e^{4/3}+\log (2)}{1+2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.16 \begin {gather*} -\frac {34-2 e^{4/3}+\log (4)}{2 (1+2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(34 - 2*E^(4/3) + 2*Log[2])/(1 + 4*x + 4*x^2),x]

[Out]

-1/2*(34 - 2*E^(4/3) + Log[4])/(1 + 2*x)

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fricas [A]  time = 0.66, size = 16, normalized size = 0.84 \begin {gather*} \frac {e^{\frac {4}{3}} - \log \relax (2) - 17}{2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(2)-2*exp(4/3)+34)/(4*x^2+4*x+1),x, algorithm="fricas")

[Out]

(e^(4/3) - log(2) - 17)/(2*x + 1)

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giac [A]  time = 0.32, size = 16, normalized size = 0.84 \begin {gather*} \frac {e^{\frac {4}{3}} - \log \relax (2) - 17}{2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(2)-2*exp(4/3)+34)/(4*x^2+4*x+1),x, algorithm="giac")

[Out]

(e^(4/3) - log(2) - 17)/(2*x + 1)

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maple [A]  time = 0.05, size = 17, normalized size = 0.89

method	result	size
norman	\(\frac {{\mathrm e}^{\frac {4}{3}}-17-\ln \relax (2)}{2 x +1}\)	\(17\)
gosper	\(-\frac {\ln \relax (2)-{\mathrm e}^{\frac {4}{3}}+17}{2 x +1}\)	\(18\)
default	\(-\frac {2 \ln \relax (2)-2 \,{\mathrm e}^{\frac {4}{3}}+34}{2 \left (2 x +1\right )}\)	\(20\)
risch	\(-\frac {\ln \relax (2)}{2 \left (\frac {1}{2}+x \right )}+\frac {{\mathrm e}^{\frac {4}{3}}}{2 x +1}-\frac {17}{2 \left (\frac {1}{2}+x \right )}\)	\(27\)
meijerg	\(\frac {2 \ln \relax (2) x}{2 x +1}-\frac {2 \,{\mathrm e}^{\frac {4}{3}} x}{2 x +1}+\frac {34 x}{2 x +1}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(2)-2*exp(4/3)+34)/(4*x^2+4*x+1),x,method=_RETURNVERBOSE)

[Out]

(exp(4/3)-17-ln(2))/(2*x+1)

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maxima [A]  time = 0.54, size = 16, normalized size = 0.84 \begin {gather*} \frac {e^{\frac {4}{3}} - \log \relax (2) - 17}{2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(2)-2*exp(4/3)+34)/(4*x^2+4*x+1),x, algorithm="maxima")

[Out]

(e^(4/3) - log(2) - 17)/(2*x + 1)

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mupad [B]  time = 0.25, size = 17, normalized size = 0.89 \begin {gather*} -\frac {\ln \relax (2)-{\mathrm {e}}^{4/3}+17}{2\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2) - 2*exp(4/3) + 34)/(4*x + 4*x^2 + 1),x)

[Out]

-(log(2) - exp(4/3) + 17)/(2*x + 1)

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sympy [A]  time = 0.07, size = 19, normalized size = 1.00 \begin {gather*} - \frac {- 2 e^{\frac {4}{3}} + 2 \log {\relax (2 )} + 34}{4 x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(2)-2*exp(4/3)+34)/(4*x**2+4*x+1),x)

[Out]

-(-2*exp(4/3) + 2*log(2) + 34)/(4*x + 2)

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