3.218 $$\int \frac{8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx$$

Optimal. Leaf size=86 $-\frac{502 x+313}{1452 \left (2 x^2+1\right )}+\frac{2843 \log \left (2 x^2+1\right )}{7986}+\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}}$

[Out]

5828/(9075*(2 - 5*x)) - (313 + 502*x)/(1452*(1 + 2*x^2)) - (251*ArcTan[Sqrt[2]*x])/(726*Sqrt[2]) + (272*Sqrt[2
]*ArcTan[Sqrt[2]*x])/1331 - (59096*Log[2 - 5*x])/99825 + (2843*Log[1 + 2*x^2])/7986

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Rubi [A]  time = 0.0978061, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 50, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {2074, 639, 203, 635, 260} $-\frac{502 x+313}{1452 \left (2 x^2+1\right )}+\frac{2843 \log \left (2 x^2+1\right )}{7986}+\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6),x]

[Out]

5828/(9075*(2 - 5*x)) - (313 + 502*x)/(1452*(1 + 2*x^2)) - (251*ArcTan[Sqrt[2]*x])/(726*Sqrt[2]) + (272*Sqrt[2
]*ArcTan[Sqrt[2]*x])/1331 - (59096*Log[2 - 5*x])/99825 + (2843*Log[1 + 2*x^2])/7986

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 260

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

Rubi steps

\begin{align*} \int \frac{8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx &=\int \left (\frac{5828}{1815 (-2+5 x)^2}-\frac{59096}{19965 (-2+5 x)}+\frac{-251+313 x}{363 \left (1+2 x^2\right )^2}+\frac{2 (816+2843 x)}{3993 \left (1+2 x^2\right )}\right ) \, dx\\ &=\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{2 \int \frac{816+2843 x}{1+2 x^2} \, dx}{3993}+\frac{1}{363} \int \frac{-251+313 x}{\left (1+2 x^2\right )^2} \, dx\\ &=\frac{5828}{9075 (2-5 x)}-\frac{313+502 x}{1452 \left (1+2 x^2\right )}-\frac{59096 \log (2-5 x)}{99825}-\frac{251}{726} \int \frac{1}{1+2 x^2} \, dx+\frac{544 \int \frac{1}{1+2 x^2} \, dx}{1331}+\frac{5686 \int \frac{x}{1+2 x^2} \, dx}{3993}\\ &=\frac{5828}{9075 (2-5 x)}-\frac{313+502 x}{1452 \left (1+2 x^2\right )}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{59096 \log (2-5 x)}{99825}+\frac{2843 \log \left (1+2 x^2\right )}{7986}\\ \end{align*}

Mathematica [A]  time = 0.0438645, size = 67, normalized size = 0.78 $\frac{-\frac{33 \left (36458 x^2+4675 x+2554\right )}{10 x^3-4 x^2+5 x-2}+142150 \log \left (2 x^2+1\right )-236384 \log (2-5 x)+12575 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{399300}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6),x]

[Out]

((-33*(2554 + 4675*x + 36458*x^2))/(-2 + 5*x - 4*x^2 + 10*x^3) + 12575*Sqrt[2]*ArcTan[Sqrt[2]*x] - 236384*Log[
2 - 5*x] + 142150*Log[1 + 2*x^2])/399300

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Maple [A]  time = 0.012, size = 54, normalized size = 0.6 \begin{align*}{\frac{1}{3993} \left ( -{\frac{2761\,x}{4}}-{\frac{3443}{8}} \right ) \left ({x}^{2}+{\frac{1}{2}} \right ) ^{-1}}+{\frac{2843\,\ln \left ( 2\,{x}^{2}+1 \right ) }{7986}}+{\frac{503\,\arctan \left ( x\sqrt{2} \right ) \sqrt{2}}{15972}}-{\frac{5828}{45375\,x-18150}}-{\frac{59096\,\ln \left ( 5\,x-2 \right ) }{99825}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x)

[Out]

1/3993*(-2761/4*x-3443/8)/(x^2+1/2)+2843/7986*ln(2*x^2+1)+503/15972*arctan(x*2^(1/2))*2^(1/2)-5828/9075/(5*x-2
)-59096/99825*ln(5*x-2)

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Maxima [A]  time = 1.42057, size = 80, normalized size = 0.93 \begin{align*} \frac{503}{15972} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - \frac{36458 \, x^{2} + 4675 \, x + 2554}{12100 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} + \frac{2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac{59096}{99825} \, \log \left (5 \, x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x, algorithm="maxima")

[Out]

503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/(10*x^3 - 4*x^2 + 5*x - 2) + 2843/79
86*log(2*x^2 + 1) - 59096/99825*log(5*x - 2)

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Fricas [A]  time = 1.9806, size = 312, normalized size = 3.63 \begin{align*} \frac{12575 \, \sqrt{2}{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\sqrt{2} x\right ) - 1203114 \, x^{2} + 142150 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (2 \, x^{2} + 1\right ) - 236384 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (5 \, x - 2\right ) - 154275 \, x - 84282}{399300 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x, algorithm="fricas")

[Out]

1/399300*(12575*sqrt(2)*(10*x^3 - 4*x^2 + 5*x - 2)*arctan(sqrt(2)*x) - 1203114*x^2 + 142150*(10*x^3 - 4*x^2 +
5*x - 2)*log(2*x^2 + 1) - 236384*(10*x^3 - 4*x^2 + 5*x - 2)*log(5*x - 2) - 154275*x - 84282)/(10*x^3 - 4*x^2 +
5*x - 2)

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Sympy [A]  time = 0.203387, size = 63, normalized size = 0.73 \begin{align*} - \frac{36458 x^{2} + 4675 x + 2554}{121000 x^{3} - 48400 x^{2} + 60500 x - 24200} - \frac{59096 \log{\left (x - \frac{2}{5} \right )}}{99825} + \frac{2843 \log{\left (x^{2} + \frac{1}{2} \right )}}{7986} + \frac{503 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{15972} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**5-7*x**3-13*x**2+8)/(100*x**6-80*x**5+116*x**4-80*x**3+41*x**2-20*x+4),x)

[Out]

-(36458*x**2 + 4675*x + 2554)/(121000*x**3 - 48400*x**2 + 60500*x - 24200) - 59096*log(x - 2/5)/99825 + 2843*l
og(x**2 + 1/2)/7986 + 503*sqrt(2)*atan(sqrt(2)*x)/15972

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Giac [A]  time = 1.0753, size = 80, normalized size = 0.93 \begin{align*} \frac{503}{15972} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - \frac{36458 \, x^{2} + 4675 \, x + 2554}{12100 \,{\left (2 \, x^{2} + 1\right )}{\left (5 \, x - 2\right )}} + \frac{2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac{59096}{99825} \, \log \left ({\left | 5 \, x - 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x, algorithm="giac")

[Out]

503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/((2*x^2 + 1)*(5*x - 2)) + 2843/7986*
log(2*x^2 + 1) - 59096/99825*log(abs(5*x - 2))