3.84.18 \(\int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+(125-75 x-110 x^2+74 x^3-15 x^4+x^5) \log (1-x^2)}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx\)

Optimal. Leaf size=23 \[ x \left (1-x-\frac {x}{(-5+x)^2}+\log \left (1-x^2\right )\right ) \]

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Rubi [B]  time = 0.84, antiderivative size = 51, normalized size of antiderivative = 2.22, number of steps used = 21, number of rules used = 8, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6741, 6725, 710, 801, 1629, 2448, 321, 206} \begin {gather*} -x^2+x \log \left (1-x^2\right )+x+\frac {10}{5-x}-\frac {25}{(5-x)^2}+\log (1-x)-\log (x+1)+2 \tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(125 - 335*x - 210*x^2 + 454*x^3 - 193*x^4 + 33*x^5 - 2*x^6 + (125 - 75*x - 110*x^2 + 74*x^3 - 15*x^4 + x^
5)*Log[1 - x^2])/(125 - 75*x - 110*x^2 + 74*x^3 - 15*x^4 + x^5),x]

[Out]

-25/(5 - x)^2 + 10/(5 - x) + x - x^2 + 2*ArcTanh[x] + Log[1 - x] - Log[1 + x] + x*Log[1 - x^2]

Rule 206

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 1629

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 6725

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{(5-x)^3 \left (1-x^2\right )} \, dx\\ &=\int \left (\frac {125}{(-5+x)^3 \left (-1+x^2\right )}-\frac {335 x}{(-5+x)^3 \left (-1+x^2\right )}-\frac {210 x^2}{(-5+x)^3 \left (-1+x^2\right )}+\frac {454 x^3}{(-5+x)^3 \left (-1+x^2\right )}-\frac {193 x^4}{(-5+x)^3 \left (-1+x^2\right )}+\frac {33 x^5}{(-5+x)^3 \left (-1+x^2\right )}-\frac {2 x^6}{(-5+x)^3 \left (-1+x^2\right )}+\log \left (1-x^2\right )\right ) \, dx\\ &=-\left (2 \int \frac {x^6}{(-5+x)^3 \left (-1+x^2\right )} \, dx\right )+33 \int \frac {x^5}{(-5+x)^3 \left (-1+x^2\right )} \, dx+125 \int \frac {1}{(-5+x)^3 \left (-1+x^2\right )} \, dx-193 \int \frac {x^4}{(-5+x)^3 \left (-1+x^2\right )} \, dx-210 \int \frac {x^2}{(-5+x)^3 \left (-1+x^2\right )} \, dx-335 \int \frac {x}{(-5+x)^3 \left (-1+x^2\right )} \, dx+454 \int \frac {x^3}{(-5+x)^3 \left (-1+x^2\right )} \, dx+\int \log \left (1-x^2\right ) \, dx\\ &=-\frac {125}{48 (5-x)^2}+x \log \left (1-x^2\right )+2 \int \frac {x^2}{1-x^2} \, dx-2 \int \left (15+\frac {15625}{24 (-5+x)^3}+\frac {146875}{288 (-5+x)^2}+\frac {521875}{3456 (-5+x)}-\frac {1}{128 (-1+x)}+x+\frac {1}{432 (1+x)}\right ) \, dx+\frac {125}{24} \int \frac {-5-x}{(-5+x)^2 \left (-1+x^2\right )} \, dx+33 \int \left (1+\frac {3125}{24 (-5+x)^3}+\frac {21875}{288 (-5+x)^2}+\frac {51875}{3456 (-5+x)}-\frac {1}{128 (-1+x)}-\frac {1}{432 (1+x)}\right ) \, dx-193 \int \left (\frac {625}{24 (-5+x)^3}+\frac {2875}{288 (-5+x)^2}+\frac {3475}{3456 (-5+x)}-\frac {1}{128 (-1+x)}+\frac {1}{432 (1+x)}\right ) \, dx-210 \int \left (\frac {25}{24 (-5+x)^3}-\frac {5}{288 (-5+x)^2}+\frac {19}{3456 (-5+x)}-\frac {1}{128 (-1+x)}+\frac {1}{432 (1+x)}\right ) \, dx-335 \int \left (\frac {5}{24 (-5+x)^3}-\frac {13}{288 (-5+x)^2}+\frac {35}{3456 (-5+x)}-\frac {1}{128 (-1+x)}-\frac {1}{432 (1+x)}\right ) \, dx+454 \int \left (\frac {125}{24 (-5+x)^3}+\frac {275}{288 (-5+x)^2}+\frac {35}{3456 (-5+x)}-\frac {1}{128 (-1+x)}-\frac {1}{432 (1+x)}\right ) \, dx\\ &=-\frac {25}{(5-x)^2}+\frac {3505}{288 (5-x)}+x-x^2+\frac {253}{128} \log (1-x)-\frac {2375 \log (5-x)}{3456}-\frac {557}{432} \log (1+x)+x \log \left (1-x^2\right )+2 \int \frac {1}{1-x^2} \, dx+\frac {125}{24} \int \left (-\frac {5}{12 (-5+x)^2}+\frac {19}{144 (-5+x)}-\frac {3}{16 (-1+x)}+\frac {1}{18 (1+x)}\right ) \, dx\\ &=-\frac {25}{(5-x)^2}+\frac {10}{5-x}+x-x^2+2 \tanh ^{-1}(x)+\log (1-x)-\log (1+x)+x \log \left (1-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 47, normalized size = 2.04 \begin {gather*} -\frac {25}{(-5+x)^2}-\frac {10}{-5+x}+x-x^2+2 \tanh ^{-1}(x)+\log (1-x)-\log (1+x)+x \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(125 - 335*x - 210*x^2 + 454*x^3 - 193*x^4 + 33*x^5 - 2*x^6 + (125 - 75*x - 110*x^2 + 74*x^3 - 15*x^
4 + x^5)*Log[1 - x^2])/(125 - 75*x - 110*x^2 + 74*x^3 - 15*x^4 + x^5),x]

[Out]

-25/(-5 + x)^2 - 10/(-5 + x) + x - x^2 + 2*ArcTanh[x] + Log[1 - x] - Log[1 + x] + x*Log[1 - x^2]

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fricas [B]  time = 0.62, size = 52, normalized size = 2.26 \begin {gather*} -\frac {x^{4} - 11 \, x^{3} + 35 \, x^{2} - {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} \log \left (-x^{2} + 1\right ) - 15 \, x - 25}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-193*x^4+454*x^3-210*x^2-335*x+125)/(x
^5-15*x^4+74*x^3-110*x^2-75*x+125),x, algorithm="fricas")

[Out]

-(x^4 - 11*x^3 + 35*x^2 - (x^3 - 10*x^2 + 25*x)*log(-x^2 + 1) - 15*x - 25)/(x^2 - 10*x + 25)

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giac [A]  time = 0.15, size = 34, normalized size = 1.48 \begin {gather*} -x^{2} + x \log \left (-x^{2} + 1\right ) + x - \frac {5 \, {\left (2 \, x - 5\right )}}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-193*x^4+454*x^3-210*x^2-335*x+125)/(x
^5-15*x^4+74*x^3-110*x^2-75*x+125),x, algorithm="giac")

[Out]

-x^2 + x*log(-x^2 + 1) + x - 5*(2*x - 5)/(x^2 - 10*x + 25)

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maple [A]  time = 0.06, size = 32, normalized size = 1.39

method	result	size
default	\(-x^{2}+x -\frac {25}{\left (x -5\right )^{2}}-\frac {10}{x -5}+x \ln \left (-x^{2}+1\right )\)	\(32\)
risch	\(x \ln \left (-x^{2}+1\right )-\frac {x^{4}-11 x^{3}+35 x^{2}-15 x -25}{x^{2}-10 x +25}\)	\(42\)
norman	\(\frac {-\frac {67 x^{2}}{2}+\ln \left (-x^{2}+1\right ) x^{3}-10 \ln \left (-x^{2}+1\right ) x^{2}+25 x \ln \left (-x^{2}+1\right )+11 x^{3}-x^{4}+\frac {125}{2}}{\left (x -5\right )^{2}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*ln(-x^2+1)-2*x^6+33*x^5-193*x^4+454*x^3-210*x^2-335*x+125)/(x^5-15*x
^4+74*x^3-110*x^2-75*x+125),x,method=_RETURNVERBOSE)

[Out]

-x^2+x-25/(x-5)^2-10/(x-5)+x*ln(-x^2+1)

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maxima [B]  time = 0.43, size = 152, normalized size = 6.61 \begin {gather*} -x^{2} + {\left (x + 1\right )} \log \left (x + 1\right ) + {\left (x - 1\right )} \log \left (-x + 1\right ) + x + \frac {3125 \, {\left (47 \, x - 205\right )}}{144 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {24125 \, {\left (23 \, x - 85\right )}}{288 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {335 \, {\left (13 \, x - 95\right )}}{288 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {5675 \, {\left (11 \, x - 25\right )}}{144 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {34375 \, {\left (7 \, x - 29\right )}}{96 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {125 \, {\left (5 \, x - 31\right )}}{288 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {175 \, {\left (x - 35\right )}}{48 \, {\left (x^{2} - 10 \, x + 25\right )}} - \log \left (x + 1\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-193*x^4+454*x^3-210*x^2-335*x+125)/(x
^5-15*x^4+74*x^3-110*x^2-75*x+125),x, algorithm="maxima")

[Out]

-x^2 + (x + 1)*log(x + 1) + (x - 1)*log(-x + 1) + x + 3125/144*(47*x - 205)/(x^2 - 10*x + 25) + 24125/288*(23*
x - 85)/(x^2 - 10*x + 25) - 335/288*(13*x - 95)/(x^2 - 10*x + 25) - 5675/144*(11*x - 25)/(x^2 - 10*x + 25) - 3
4375/96*(7*x - 29)/(x^2 - 10*x + 25) + 125/288*(5*x - 31)/(x^2 - 10*x + 25) - 175/48*(x - 35)/(x^2 - 10*x + 25
) - log(x + 1) + log(x - 1)

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mupad [B]  time = 5.27, size = 34, normalized size = 1.48 \begin {gather*} x-\frac {10\,x-25}{x^2-10\,x+25}-x^2+x\,\ln \left (1-x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((335*x + log(1 - x^2)*(75*x + 110*x^2 - 74*x^3 + 15*x^4 - x^5 - 125) + 210*x^2 - 454*x^3 + 193*x^4 - 33*x^
5 + 2*x^6 - 125)/(75*x + 110*x^2 - 74*x^3 + 15*x^4 - x^5 - 125),x)

[Out]

x - (10*x - 25)/(x^2 - 10*x + 25) - x^2 + x*log(1 - x^2)

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sympy [A]  time = 0.18, size = 26, normalized size = 1.13 \begin {gather*} - x^{2} + x \log {\left (1 - x^{2} \right )} + x - \frac {10 x - 25}{x^{2} - 10 x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5-15*x**4+74*x**3-110*x**2-75*x+125)*ln(-x**2+1)-2*x**6+33*x**5-193*x**4+454*x**3-210*x**2-335*
x+125)/(x**5-15*x**4+74*x**3-110*x**2-75*x+125),x)

[Out]

-x**2 + x*log(1 - x**2) + x - (10*x - 25)/(x**2 - 10*x + 25)

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