3.44.84 \(\int \frac {-12-34 x+12 x^2+e (-1+8 x)+(-16-x+4 x^2) \log (-64-4 x+16 x^2)}{-16-x+4 x^2} \, dx\)

Optimal. Leaf size=20 \[ x+(-4+e+x) \log \left (4 \left (-16-x+4 x^2\right )\right ) \]

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Rubi [B]  time = 0.21, antiderivative size = 117, normalized size of antiderivative = 5.85, number of steps used = 12, number of rules used = 6, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6728, 1657, 632, 31, 2523, 773} \begin {gather*} x \log \left (16 x^2-4 x-64\right )+x-\frac {1}{8} \left (31+\sqrt {257}-8 e\right ) \log \left (-8 x-\sqrt {257}+1\right )-\frac {1}{8} \left (1-\sqrt {257}\right ) \log \left (-8 x-\sqrt {257}+1\right )-\frac {1}{8} \left (31-\sqrt {257}-8 e\right ) \log \left (-8 x+\sqrt {257}+1\right )-\frac {1}{8} \left (1+\sqrt {257}\right ) \log \left (-8 x+\sqrt {257}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 - 34*x + 12*x^2 + E*(-1 + 8*x) + (-16 - x + 4*x^2)*Log[-64 - 4*x + 16*x^2])/(-16 - x + 4*x^2),x]

[Out]

x - ((1 - Sqrt[257])*Log[1 - Sqrt[257] - 8*x])/8 - ((31 + Sqrt[257] - 8*E)*Log[1 - Sqrt[257] - 8*x])/8 - ((1 +
Sqrt[257])*Log[1 + Sqrt[257] - 8*x])/8 - ((31 - Sqrt[257] - 8*E)*Log[1 + Sqrt[257] - 8*x])/8 + x*Log[-64 - 4*
x + 16*x^2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 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]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {12+e+2 (17-4 e) x-12 x^2}{16+x-4 x^2}+\log \left (-64-4 x+16 x^2\right )\right ) \, dx\\ &=\int \frac {12+e+2 (17-4 e) x-12 x^2}{16+x-4 x^2} \, dx+\int \log \left (-64-4 x+16 x^2\right ) \, dx\\ &=x \log \left (-64-4 x+16 x^2\right )-\int \frac {(1-8 x) x}{16+x-4 x^2} \, dx+\int \left (3-\frac {36-e-(31-8 e) x}{16+x-4 x^2}\right ) \, dx\\ &=x+x \log \left (-64-4 x+16 x^2\right )+\frac {1}{4} \int \frac {128+4 x}{16+x-4 x^2} \, dx-\int \frac {36-e-(31-8 e) x}{16+x-4 x^2} \, dx\\ &=x+x \log \left (-64-4 x+16 x^2\right )+\frac {1}{2} \left (1-\sqrt {257}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {257}}{2}-4 x} \, dx+\frac {1}{2} \left (1+\sqrt {257}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {257}}{2}-4 x} \, dx+\frac {1}{2} \left (31-\sqrt {257}-8 e\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {257}}{2}-4 x} \, dx-\frac {1}{2} \left (-31-\sqrt {257}+8 e\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {257}}{2}-4 x} \, dx\\ &=x-\frac {1}{8} \left (1-\sqrt {257}\right ) \log \left (1-\sqrt {257}-8 x\right )-\frac {1}{8} \left (31+\sqrt {257}-8 e\right ) \log \left (1-\sqrt {257}-8 x\right )-\frac {1}{8} \left (1+\sqrt {257}\right ) \log \left (1+\sqrt {257}-8 x\right )-\frac {1}{8} \left (31-\sqrt {257}-8 e\right ) \log \left (1+\sqrt {257}-8 x\right )+x \log \left (-64-4 x+16 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 71, normalized size = 3.55 \begin {gather*} x-4 \log \left (1-\sqrt {257}-8 x\right )+e \log \left (1-\sqrt {257}-8 x\right )-4 \log \left (1+\sqrt {257}-8 x\right )+e \log \left (1+\sqrt {257}-8 x\right )+x \log \left (-64-4 x+16 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 - 34*x + 12*x^2 + E*(-1 + 8*x) + (-16 - x + 4*x^2)*Log[-64 - 4*x + 16*x^2])/(-16 - x + 4*x^2),x
]

[Out]

x - 4*Log[1 - Sqrt[257] - 8*x] + E*Log[1 - Sqrt[257] - 8*x] - 4*Log[1 + Sqrt[257] - 8*x] + E*Log[1 + Sqrt[257]
- 8*x] + x*Log[-64 - 4*x + 16*x^2]

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fricas [A]  time = 0.80, size = 19, normalized size = 0.95 \begin {gather*} {\left (x + e - 4\right )} \log \left (16 \, x^{2} - 4 \, x - 64\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2-x-16)*log(16*x^2-4*x-64)+(8*x-1)*exp(1)+12*x^2-34*x-12)/(4*x^2-x-16),x, algorithm="fricas")

[Out]

(x + e - 4)*log(16*x^2 - 4*x - 64) + x

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giac [B]  time = 2.21, size = 42, normalized size = 2.10 \begin {gather*} x \log \left (16 \, x^{2} - 4 \, x - 64\right ) + e \log \left (4 \, x^{2} - x - 16\right ) + x - 4 \, \log \left (4 \, x^{2} - x - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2-x-16)*log(16*x^2-4*x-64)+(8*x-1)*exp(1)+12*x^2-34*x-12)/(4*x^2-x-16),x, algorithm="giac")

[Out]

x*log(16*x^2 - 4*x - 64) + e*log(4*x^2 - x - 16) + x - 4*log(4*x^2 - x - 16)

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maple [A]  time = 0.20, size = 32, normalized size = 1.60

method	result	size
norman	\(x +\left ({\mathrm e}-4\right ) \ln \left (16 x^{2}-4 x -64\right )+\ln \left (16 x^{2}-4 x -64\right ) x\)	\(32\)
risch	\(\ln \left (16 x^{2}-4 x -64\right ) x +\ln \left (4 x^{2}-x -16\right ) {\mathrm e}-4 \ln \left (4 x^{2}-x -16\right )+x\)	\(43\)
default	\(x -4 \ln \left (4 x^{2}-x -16\right )+\ln \left (4 x^{2}-x -16\right ) {\mathrm e}+2 x \ln \relax (2)+x \ln \left (4 x^{2}-x -16\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2-x-16)*ln(16*x^2-4*x-64)+(8*x-1)*exp(1)+12*x^2-34*x-12)/(4*x^2-x-16),x,method=_RETURNVERBOSE)

[Out]

x+(exp(1)-4)*ln(16*x^2-4*x-64)+ln(16*x^2-4*x-64)*x

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maxima [B]  time = 0.51, size = 115, normalized size = 5.75 \begin {gather*} -\frac {1}{257} \, \sqrt {257} e \log \left (\frac {8 \, x - \sqrt {257} - 1}{8 \, x + \sqrt {257} - 1}\right ) + 2 \, x {\left (\log \relax (2) - 1\right )} + \frac {1}{257} \, {\left (\sqrt {257} \log \left (\frac {8 \, x - \sqrt {257} - 1}{8 \, x + \sqrt {257} - 1}\right ) + 257 \, \log \left (4 \, x^{2} - x - 16\right )\right )} e + \frac {1}{8} \, {\left (8 \, x - 1\right )} \log \left (4 \, x^{2} - x - 16\right ) + 3 \, x - \frac {31}{8} \, \log \left (4 \, x^{2} - x - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2-x-16)*log(16*x^2-4*x-64)+(8*x-1)*exp(1)+12*x^2-34*x-12)/(4*x^2-x-16),x, algorithm="maxima")

[Out]

-1/257*sqrt(257)*e*log((8*x - sqrt(257) - 1)/(8*x + sqrt(257) - 1)) + 2*x*(log(2) - 1) + 1/257*(sqrt(257)*log(
(8*x - sqrt(257) - 1)/(8*x + sqrt(257) - 1)) + 257*log(4*x^2 - x - 16))*e + 1/8*(8*x - 1)*log(4*x^2 - x - 16)
+ 3*x - 31/8*log(4*x^2 - x - 16)

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mupad [B]  time = 3.20, size = 31, normalized size = 1.55 \begin {gather*} x+x\,\ln \left (16\,x^2-4\,x-64\right )+\ln \left (4\,x^2-x-16\right )\,\left (\mathrm {e}-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((34*x + log(16*x^2 - 4*x - 64)*(x - 4*x^2 + 16) - 12*x^2 - exp(1)*(8*x - 1) + 12)/(x - 4*x^2 + 16),x)

[Out]

x + x*log(16*x^2 - 4*x - 64) + log(4*x^2 - x - 16)*(exp(1) - 4)

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sympy [A]  time = 0.27, size = 29, normalized size = 1.45 \begin {gather*} x \log {\left (16 x^{2} - 4 x - 64 \right )} + x + \left (-4 + e\right ) \log {\left (4 x^{2} - x - 16 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2-x-16)*ln(16*x**2-4*x-64)+(8*x-1)*exp(1)+12*x**2-34*x-12)/(4*x**2-x-16),x)

[Out]

x*log(16*x**2 - 4*x - 64) + x + (-4 + E)*log(4*x**2 - x - 16)

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