3.15.4 \(\int \frac {-7+(-2 x-x^2) \log (2)+(1+x) \log (\log (375))}{-7-x^2 \log (2)+x \log (\log (375))} \, dx\)

Optimal. Leaf size=17 \[ x+\log (7+x (x \log (2)-\log (\log (375)))) \]

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1657, 628} \begin {gather*} \log \left (x^2 \log (2)-x \log (\log (375))+7\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-7 + (-2*x - x^2)*Log[2] + (1 + x)*Log[Log[375]])/(-7 - x^2*Log[2] + x*Log[Log[375]]),x]

[Out]

x + Log[7 + x^2*Log[2] - x*Log[Log[375]]]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {-2 x \log (2)+\log (\log (375))}{-7-x^2 \log (2)+x \log (\log (375))}\right ) \, dx\\ &=x+\int \frac {-2 x \log (2)+\log (\log (375))}{-7-x^2 \log (2)+x \log (\log (375))} \, dx\\ &=x+\log \left (7+x^2 \log (2)-x \log (\log (375))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} x+\log \left (7+x^2 \log (2)-x \log (\log (375))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7 + (-2*x - x^2)*Log[2] + (1 + x)*Log[Log[375]])/(-7 - x^2*Log[2] + x*Log[Log[375]]),x]

[Out]

x + Log[7 + x^2*Log[2] - x*Log[Log[375]]]

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fricas [A]  time = 0.59, size = 17, normalized size = 1.00 \begin {gather*} x + \log \left (x^{2} \log \relax (2) - x \log \left (\log \left (375\right )\right ) + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*log(log(375))+(-x^2-2*x)*log(2)-7)/(x*log(log(375))-x^2*log(2)-7),x, algorithm="fricas")

[Out]

x + log(x^2*log(2) - x*log(log(375)) + 7)

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giac [A]  time = 0.41, size = 17, normalized size = 1.00 \begin {gather*} x + \log \left (x^{2} \log \relax (2) - x \log \left (\log \left (375\right )\right ) + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*log(log(375))+(-x^2-2*x)*log(2)-7)/(x*log(log(375))-x^2*log(2)-7),x, algorithm="giac")

[Out]

x + log(x^2*log(2) - x*log(log(375)) + 7)

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maple [A]  time = 0.72, size = 18, normalized size = 1.06

method	result	size
default	\(x +\ln \left (x^{2} \ln \relax (2)-x \ln \left (\ln \left (375\right )\right )+7\right )\)	\(18\)
norman	\(x +\ln \left (x^{2} \ln \relax (2)-x \ln \left (\ln \left (375\right )\right )+7\right )\)	\(18\)
risch	\(x +\ln \left (x^{2} \ln \relax (2)-x \ln \left (\ln \relax (3)+3 \ln \relax (5)\right )+7\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x+1)*ln(ln(375))+(-x^2-2*x)*ln(2)-7)/(x*ln(ln(375))-x^2*ln(2)-7),x,method=_RETURNVERBOSE)

[Out]

x+ln(x^2*ln(2)-x*ln(ln(375))+7)

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maxima [A]  time = 0.45, size = 17, normalized size = 1.00 \begin {gather*} x + \log \left (x^{2} \log \relax (2) - x \log \left (\log \left (375\right )\right ) + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*log(log(375))+(-x^2-2*x)*log(2)-7)/(x*log(log(375))-x^2*log(2)-7),x, algorithm="maxima")

[Out]

x + log(x^2*log(2) - x*log(log(375)) + 7)

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mupad [B]  time = 0.13, size = 17, normalized size = 1.00 \begin {gather*} x+\ln \left (\ln \relax (2)\,x^2-\ln \left (\ln \left (375\right )\right )\,x+7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(2*x + x^2) - log(log(375))*(x + 1) + 7)/(x^2*log(2) - x*log(log(375)) + 7),x)

[Out]

x + log(x^2*log(2) - x*log(log(375)) + 7)

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sympy [A]  time = 0.20, size = 17, normalized size = 1.00 \begin {gather*} x + \log {\left (x^{2} \log {\relax (2 )} - x \log {\left (\log {\left (375 \right )} \right )} + 7 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*ln(ln(375))+(-x**2-2*x)*ln(2)-7)/(x*ln(ln(375))-x**2*ln(2)-7),x)

[Out]

x + log(x**2*log(2) - x*log(log(375)) + 7)

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