3.86.32 \(\int \frac {-576 x^2+192 x^3+(-144+96 x-144 x^2-624 x^3+240 x^4) \log (1+x^2)+(120 x^2+120 x^4) \log ^2(1+x^2)+(225 x^2+225 x^4) \log ^3(1+x^2)}{(144 x^2-96 x^3+160 x^4-96 x^5+16 x^6) \log ^3(1+x^2)} \, dx\)

Optimal. Leaf size=32 \[ \frac {3 \left (x+\frac {1}{4} \left (x+\frac {4}{\log \left (1+x^2\right )}\right )\right )^2}{(3-x) x} \]

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Rubi [F]  time = 2.33, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{\left (144 x^2-96 x^3+160 x^4-96 x^5+16 x^6\right ) \log ^3\left (1+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-576*x^2 + 192*x^3 + (-144 + 96*x - 144*x^2 - 624*x^3 + 240*x^4)*Log[1 + x^2] + (120*x^2 + 120*x^4)*Log[1
+ x^2]^2 + (225*x^2 + 225*x^4)*Log[1 + x^2]^3)/((144*x^2 - 96*x^3 + 160*x^4 - 96*x^5 + 16*x^6)*Log[1 + x^2]^3
),x]

[Out]

225/(16*(3 - x)) + 3/(10*Log[1 + x^2]^2) + 9/(4*Log[1 + x^2]) + (6*Defer[Int][1/((-3 + x)*Log[1 + x^2]^3), x])
/5 - (18*Defer[Int][1/((1 + x^2)*Log[1 + x^2]^3), x])/5 + Defer[Int][1/((-3 + x)^2*Log[1 + x^2]^2), x] + (9*De
fer[Int][1/((-3 + x)*Log[1 + x^2]^2), x])/2 - Defer[Int][1/(x^2*Log[1 + x^2]^2), x] + (3*Defer[Int][1/((1 + x^
2)*Log[1 + x^2]^2), x])/2 + (15*Defer[Int][1/((-3 + x)^2*Log[1 + x^2]), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{16 (3-x)^2 x^2 \left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx\\ &=\frac {1}{16} \int \frac {-576 x^2+192 x^3+\left (-144+96 x-144 x^2-624 x^3+240 x^4\right ) \log \left (1+x^2\right )+\left (120 x^2+120 x^4\right ) \log ^2\left (1+x^2\right )+\left (225 x^2+225 x^4\right ) \log ^3\left (1+x^2\right )}{(3-x)^2 x^2 \left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx\\ &=\frac {1}{16} \int \left (\frac {225}{(-3+x)^2}+\frac {192}{(-3+x) \left (1+x^2\right ) \log ^3\left (1+x^2\right )}+\frac {48 \left (-3+2 x-3 x^2-13 x^3+5 x^4\right )}{(-3+x)^2 x^2 \left (1+x^2\right ) \log ^2\left (1+x^2\right )}+\frac {120}{(-3+x)^2 \log \left (1+x^2\right )}\right ) \, dx\\ &=\frac {225}{16 (3-x)}+3 \int \frac {-3+2 x-3 x^2-13 x^3+5 x^4}{(-3+x)^2 x^2 \left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+12 \int \frac {1}{(-3+x) \left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}+3 \int \left (\frac {1}{3 (-3+x)^2 \log ^2\left (1+x^2\right )}+\frac {3}{2 (-3+x) \log ^2\left (1+x^2\right )}-\frac {1}{3 x^2 \log ^2\left (1+x^2\right )}+\frac {1-3 x}{2 \left (1+x^2\right ) \log ^2\left (1+x^2\right )}\right ) \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+12 \int \left (\frac {1}{10 (-3+x) \log ^3\left (1+x^2\right )}+\frac {-3-x}{10 \left (1+x^2\right ) \log ^3\left (1+x^2\right )}\right ) \, dx\\ &=\frac {225}{16 (3-x)}+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx+\frac {6}{5} \int \frac {-3-x}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \frac {1-3 x}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}+\frac {6}{5} \int \left (-\frac {3}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )}-\frac {x}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )}\right ) \, dx+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \left (\frac {1}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )}-\frac {3 x}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )}\right ) \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx-\frac {6}{5} \int \frac {x}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \frac {1}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx-\frac {18}{5} \int \frac {1}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx-\frac {9}{2} \int \frac {x}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}-\frac {3}{5} \operatorname {Subst}\left (\int \frac {1}{(1+x) \log ^3(1+x)} \, dx,x,x^2\right )+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \frac {1}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{(1+x) \log ^2(1+x)} \, dx,x,x^2\right )-\frac {18}{5} \int \frac {1}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}-\frac {3}{5} \operatorname {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,1+x^2\right )+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \frac {1}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,1+x^2\right )-\frac {18}{5} \int \frac {1}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}-\frac {3}{5} \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (1+x^2\right )\right )+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \frac {1}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (1+x^2\right )\right )-\frac {18}{5} \int \frac {1}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ &=\frac {225}{16 (3-x)}+\frac {3}{10 \log ^2\left (1+x^2\right )}+\frac {9}{4 \log \left (1+x^2\right )}+\frac {6}{5} \int \frac {1}{(-3+x) \log ^3\left (1+x^2\right )} \, dx+\frac {3}{2} \int \frac {1}{\left (1+x^2\right ) \log ^2\left (1+x^2\right )} \, dx-\frac {18}{5} \int \frac {1}{\left (1+x^2\right ) \log ^3\left (1+x^2\right )} \, dx+\frac {9}{2} \int \frac {1}{(-3+x) \log ^2\left (1+x^2\right )} \, dx+\frac {15}{2} \int \frac {1}{(-3+x)^2 \log \left (1+x^2\right )} \, dx+\int \frac {1}{(-3+x)^2 \log ^2\left (1+x^2\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (1+x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 45, normalized size = 1.41 \begin {gather*} \frac {3}{16} \left (-\frac {75}{-3+x}-\frac {16}{(-3+x) x \log ^2\left (1+x^2\right )}-\frac {40}{(-3+x) \log \left (1+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-576*x^2 + 192*x^3 + (-144 + 96*x - 144*x^2 - 624*x^3 + 240*x^4)*Log[1 + x^2] + (120*x^2 + 120*x^4)
*Log[1 + x^2]^2 + (225*x^2 + 225*x^4)*Log[1 + x^2]^3)/((144*x^2 - 96*x^3 + 160*x^4 - 96*x^5 + 16*x^6)*Log[1 +
x^2]^3),x]

[Out]

(3*(-75/(-3 + x) - 16/((-3 + x)*x*Log[1 + x^2]^2) - 40/((-3 + x)*Log[1 + x^2])))/16

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fricas [A]  time = 0.52, size = 41, normalized size = 1.28 \begin {gather*} -\frac {3 \, {\left (75 \, x \log \left (x^{2} + 1\right )^{2} + 40 \, x \log \left (x^{2} + 1\right ) + 16\right )}}{16 \, {\left (x^{2} - 3 \, x\right )} \log \left (x^{2} + 1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+(240*x^4-624*x^3-144*x^2+96*x-144)*lo
g(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x^5+160*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x, algorithm="fricas")

[Out]

-3/16*(75*x*log(x^2 + 1)^2 + 40*x*log(x^2 + 1) + 16)/((x^2 - 3*x)*log(x^2 + 1)^2)

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giac [A]  time = 0.22, size = 47, normalized size = 1.47 \begin {gather*} -\frac {3 \, {\left (5 \, x \log \left (x^{2} + 1\right ) + 2\right )}}{2 \, {\left (x^{2} \log \left (x^{2} + 1\right )^{2} - 3 \, x \log \left (x^{2} + 1\right )^{2}\right )}} - \frac {225}{16 \, {\left (x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+(240*x^4-624*x^3-144*x^2+96*x-144)*lo
g(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x^5+160*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x, algorithm="giac")

[Out]

-3/2*(5*x*log(x^2 + 1) + 2)/(x^2*log(x^2 + 1)^2 - 3*x*log(x^2 + 1)^2) - 225/16/(x - 3)

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maple [A]  time = 0.05, size = 38, normalized size = 1.19

method	result	size
risch	\(-\frac {225}{16 \left (x -3\right )}-\frac {3 \left (5 x \ln \left (x^{2}+1\right )+2\right )}{2 x \left (x -3\right ) \ln \left (x^{2}+1\right )^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((225*x^4+225*x^2)*ln(x^2+1)^3+(120*x^4+120*x^2)*ln(x^2+1)^2+(240*x^4-624*x^3-144*x^2+96*x-144)*ln(x^2+1)+
192*x^3-576*x^2)/(16*x^6-96*x^5+160*x^4-96*x^3+144*x^2)/ln(x^2+1)^3,x,method=_RETURNVERBOSE)

[Out]

-225/16/(x-3)-3/2/x*(5*x*ln(x^2+1)+2)/(x-3)/ln(x^2+1)^2

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maxima [A]  time = 0.49, size = 41, normalized size = 1.28 \begin {gather*} -\frac {3 \, {\left (75 \, x \log \left (x^{2} + 1\right )^{2} + 40 \, x \log \left (x^{2} + 1\right ) + 16\right )}}{16 \, {\left (x^{2} - 3 \, x\right )} \log \left (x^{2} + 1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((225*x^4+225*x^2)*log(x^2+1)^3+(120*x^4+120*x^2)*log(x^2+1)^2+(240*x^4-624*x^3-144*x^2+96*x-144)*lo
g(x^2+1)+192*x^3-576*x^2)/(16*x^6-96*x^5+160*x^4-96*x^3+144*x^2)/log(x^2+1)^3,x, algorithm="maxima")

[Out]

-3/16*(75*x*log(x^2 + 1)^2 + 40*x*log(x^2 + 1) + 16)/((x^2 - 3*x)*log(x^2 + 1)^2)

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mupad [B]  time = 5.57, size = 313, normalized size = 9.78 \begin {gather*} \frac {\frac {165\,x^7}{16}-\frac {573\,x^6}{8}+\frac {1911\,x^5}{16}+\frac {153\,x^4}{8}-\frac {27\,x^3}{2}+\frac {33\,x^2}{2}-\frac {81\,x}{8}+\frac {81}{8}}{-x^8+9\,x^7-27\,x^6+27\,x^5}-\frac {\frac {3\,\left (5\,x^4-17\,x^3+3\,x^2-2\,x+3\right )}{4\,x^3\,{\left (x-3\right )}^2}+\frac {3\,\ln \left (x^2+1\right )\,\left (x^2+1\right )\,\left (5\,x^5-19\,x^4+9\,x^3-17\,x^2+27\,x-27\right )}{8\,x^5\,{\left (x-3\right )}^3}-\frac {15\,{\ln \left (x^2+1\right )}^2\,\left (x^2+1\right )\,\left (x^3+3\,x^2+3\,x-3\right )}{16\,x^3\,{\left (x-3\right )}^3}}{\ln \left (x^2+1\right )}-\frac {\frac {3}{x\,\left (x-3\right )}+\frac {15\,{\ln \left (x^2+1\right )}^2\,\left (x^2+1\right )}{8\,x\,{\left (x-3\right )}^2}-\frac {3\,\ln \left (x^2+1\right )\,\left (-5\,x^4+13\,x^3+3\,x^2-2\,x+3\right )}{4\,x^3\,{\left (x-3\right )}^2}}{{\ln \left (x^2+1\right )}^2}+\frac {\ln \left (x^2+1\right )\,\left (\frac {15\,x^5}{16}+\frac {45\,x^4}{16}+\frac {15\,x^3}{4}+\frac {45\,x}{16}-\frac {45}{16}\right )}{-x^6+9\,x^5-27\,x^4+27\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 + 1)^2*(120*x^2 + 120*x^4) + log(x^2 + 1)^3*(225*x^2 + 225*x^4) - 576*x^2 + 192*x^3 - log(x^2 + 1
)*(144*x^2 - 96*x + 624*x^3 - 240*x^4 + 144))/(log(x^2 + 1)^3*(144*x^2 - 96*x^3 + 160*x^4 - 96*x^5 + 16*x^6)),
x)

[Out]

((33*x^2)/2 - (81*x)/8 - (27*x^3)/2 + (153*x^4)/8 + (1911*x^5)/16 - (573*x^6)/8 + (165*x^7)/16 + 81/8)/(27*x^5
- 27*x^6 + 9*x^7 - x^8) - ((3*(3*x^2 - 2*x - 17*x^3 + 5*x^4 + 3))/(4*x^3*(x - 3)^2) + (3*log(x^2 + 1)*(x^2 +
1)*(27*x - 17*x^2 + 9*x^3 - 19*x^4 + 5*x^5 - 27))/(8*x^5*(x - 3)^3) - (15*log(x^2 + 1)^2*(x^2 + 1)*(3*x + 3*x^
2 + x^3 - 3))/(16*x^3*(x - 3)^3))/log(x^2 + 1) - (3/(x*(x - 3)) + (15*log(x^2 + 1)^2*(x^2 + 1))/(8*x*(x - 3)^2
) - (3*log(x^2 + 1)*(3*x^2 - 2*x + 13*x^3 - 5*x^4 + 3))/(4*x^3*(x - 3)^2))/log(x^2 + 1)^2 + (log(x^2 + 1)*((45
*x)/16 + (15*x^3)/4 + (45*x^4)/16 + (15*x^5)/16 - 45/16))/(27*x^3 - 27*x^4 + 9*x^5 - x^6)

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sympy [A]  time = 0.21, size = 36, normalized size = 1.12 \begin {gather*} \frac {- 15 x \log {\left (x^{2} + 1 \right )} - 6}{\left (2 x^{2} - 6 x\right ) \log {\left (x^{2} + 1 \right )}^{2}} - \frac {225}{16 x - 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((225*x**4+225*x**2)*ln(x**2+1)**3+(120*x**4+120*x**2)*ln(x**2+1)**2+(240*x**4-624*x**3-144*x**2+96*
x-144)*ln(x**2+1)+192*x**3-576*x**2)/(16*x**6-96*x**5+160*x**4-96*x**3+144*x**2)/ln(x**2+1)**3,x)

[Out]

(-15*x*log(x**2 + 1) - 6)/((2*x**2 - 6*x)*log(x**2 + 1)**2) - 225/(16*x - 48)

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