3.73.3 \(\int \frac {-576 x+(-576-576 x) \log (1+x)+(180 x^2+180 x^3) \log ^2(1+x)}{2304+2304 x+(-1056 x+384 x^2+1440 x^3) \log (1+x)+(121 x^2-209 x^3-105 x^4+225 x^5) \log ^2(1+x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {x}{\frac {2 x}{3}-x \left (\frac {1}{4} (-1+x)+x\right )-\frac {4}{\log (1+x)}} \]

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Rubi [F]  time = 3.57, 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+(-576-576 x) \log (1+x)+\left (180 x^2+180 x^3\right ) \log ^2(1+x)}{2304+2304 x+\left (-1056 x+384 x^2+1440 x^3\right ) \log (1+x)+\left (121 x^2-209 x^3-105 x^4+225 x^5\right ) \log ^2(1+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-576*x + (-576 - 576*x)*Log[1 + x] + (180*x^2 + 180*x^3)*Log[1 + x]^2)/(2304 + 2304*x + (-1056*x + 384*x^
2 + 1440*x^3)*Log[1 + x] + (121*x^2 - 209*x^3 - 105*x^4 + 225*x^5)*Log[1 + x]^2),x]

[Out]

12/(11 - 15*x) - 576*Defer[Int][(48 - 11*x*Log[1 + x] + 15*x^2*Log[1 + x])^(-2), x] - (27648*Defer[Int][1/(x*(
48 - 11*x*Log[1 + x] + 15*x^2*Log[1 + x])^2), x])/11 + 576*Defer[Int][1/((1 + x)*(48 - 11*x*Log[1 + x] + 15*x^
2*Log[1 + x])^2), x] + 414720*Defer[Int][1/((-11 + 15*x)^2*(48 - 11*x*Log[1 + x] + 15*x^2*Log[1 + x])^2), x] +
(414720*Defer[Int][1/((-11 + 15*x)*(48 - 11*x*Log[1 + x] + 15*x^2*Log[1 + x])^2), x])/11 + (576*Defer[Int][1/
(x*(48 - 11*x*Log[1 + x] + 15*x^2*Log[1 + x])), x])/11 - 17280*Defer[Int][1/((-11 + 15*x)^2*(48 - 11*x*Log[1 +
x] + 15*x^2*Log[1 + x])), x] - (8640*Defer[Int][1/((-11 + 15*x)*(48 - 11*x*Log[1 + x] + 15*x^2*Log[1 + x])),
x])/11

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36 \left (-16 x-16 (1+x) \log (1+x)+5 x^2 (1+x) \log ^2(1+x)\right )}{(1+x) (48+x (-11+15 x) \log (1+x))^2} \, dx\\ &=36 \int \frac {-16 x-16 (1+x) \log (1+x)+5 x^2 (1+x) \log ^2(1+x)}{(1+x) (48+x (-11+15 x) \log (1+x))^2} \, dx\\ &=36 \int \left (\frac {5}{(-11+15 x)^2}-\frac {16 \left (528-912 x-1319 x^2-330 x^3+225 x^4\right )}{x (1+x) (-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2}-\frac {16 (-11+45 x)}{x (-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )}\right ) \, dx\\ &=\frac {12}{11-15 x}-576 \int \frac {528-912 x-1319 x^2-330 x^3+225 x^4}{x (1+x) (-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2} \, dx-576 \int \frac {-11+45 x}{x (-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )} \, dx\\ &=\frac {12}{11-15 x}-576 \int \left (\frac {1}{\left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2}+\frac {48}{11 x \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2}-\frac {1}{(1+x) \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2}-\frac {720}{(-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2}-\frac {720}{11 (-11+15 x) \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2}\right ) \, dx-576 \int \left (-\frac {1}{11 x \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )}+\frac {30}{(-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )}+\frac {15}{11 (-11+15 x) \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )}\right ) \, dx\\ &=\frac {12}{11-15 x}+\frac {576}{11} \int \frac {1}{x \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )} \, dx-576 \int \frac {1}{\left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2} \, dx+576 \int \frac {1}{(1+x) \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2} \, dx-\frac {8640}{11} \int \frac {1}{(-11+15 x) \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )} \, dx-\frac {27648}{11} \int \frac {1}{x \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2} \, dx-17280 \int \frac {1}{(-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )} \, dx+\frac {414720}{11} \int \frac {1}{(-11+15 x) \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2} \, dx+414720 \int \frac {1}{(-11+15 x)^2 \left (48-11 x \log (1+x)+15 x^2 \log (1+x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.08, size = 23, normalized size = 0.77 \begin {gather*} -\frac {36 x \log (1+x)}{144+3 x (-11+15 x) \log (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-576*x + (-576 - 576*x)*Log[1 + x] + (180*x^2 + 180*x^3)*Log[1 + x]^2)/(2304 + 2304*x + (-1056*x +
384*x^2 + 1440*x^3)*Log[1 + x] + (121*x^2 - 209*x^3 - 105*x^4 + 225*x^5)*Log[1 + x]^2),x]

[Out]

(-36*x*Log[1 + x])/(144 + 3*x*(-11 + 15*x)*Log[1 + x])

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fricas [A]  time = 0.74, size = 25, normalized size = 0.83 \begin {gather*} -\frac {12 \, x \log \left (x + 1\right )}{{\left (15 \, x^{2} - 11 \, x\right )} \log \left (x + 1\right ) + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((180*x^3+180*x^2)*log(x+1)^2+(-576*x-576)*log(x+1)-576*x)/((225*x^5-105*x^4-209*x^3+121*x^2)*log(x+
1)^2+(1440*x^3+384*x^2-1056*x)*log(x+1)+2304*x+2304),x, algorithm="fricas")

[Out]

-12*x*log(x + 1)/((15*x^2 - 11*x)*log(x + 1) + 48)

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giac [A]  time = 0.21, size = 44, normalized size = 1.47 \begin {gather*} \frac {576}{225 \, x^{3} \log \left (x + 1\right ) - 330 \, x^{2} \log \left (x + 1\right ) + 121 \, x \log \left (x + 1\right ) + 720 \, x - 528} - \frac {12}{15 \, x - 11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((180*x^3+180*x^2)*log(x+1)^2+(-576*x-576)*log(x+1)-576*x)/((225*x^5-105*x^4-209*x^3+121*x^2)*log(x+
1)^2+(1440*x^3+384*x^2-1056*x)*log(x+1)+2304*x+2304),x, algorithm="giac")

[Out]

576/(225*x^3*log(x + 1) - 330*x^2*log(x + 1) + 121*x*log(x + 1) + 720*x - 528) - 12/(15*x - 11)

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maple [A]  time = 0.06, size = 33, normalized size = 1.10

method	result	size
norman	\(\frac {-\frac {180 \ln \left (x +1\right ) x^{2}}{11}-\frac {576}{11}}{15 \ln \left (x +1\right ) x^{2}-11 \ln \left (x +1\right ) x +48}\)	\(33\)
risch	\(-\frac {12}{15 x -11}+\frac {576}{\left (15 x -11\right ) \left (15 \ln \left (x +1\right ) x^{2}-11 \ln \left (x +1\right ) x +48\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((180*x^3+180*x^2)*ln(x+1)^2+(-576*x-576)*ln(x+1)-576*x)/((225*x^5-105*x^4-209*x^3+121*x^2)*ln(x+1)^2+(144
0*x^3+384*x^2-1056*x)*ln(x+1)+2304*x+2304),x,method=_RETURNVERBOSE)

[Out]

(-180/11*ln(x+1)*x^2-576/11)/(15*ln(x+1)*x^2-11*ln(x+1)*x+48)

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maxima [A]  time = 0.41, size = 25, normalized size = 0.83 \begin {gather*} -\frac {12 \, x \log \left (x + 1\right )}{{\left (15 \, x^{2} - 11 \, x\right )} \log \left (x + 1\right ) + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((180*x^3+180*x^2)*log(x+1)^2+(-576*x-576)*log(x+1)-576*x)/((225*x^5-105*x^4-209*x^3+121*x^2)*log(x+
1)^2+(1440*x^3+384*x^2-1056*x)*log(x+1)+2304*x+2304),x, algorithm="maxima")

[Out]

-12*x*log(x + 1)/((15*x^2 - 11*x)*log(x + 1) + 48)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {\left (-180\,x^3-180\,x^2\right )\,{\ln \left (x+1\right )}^2+\left (576\,x+576\right )\,\ln \left (x+1\right )+576\,x}{\left (225\,x^5-105\,x^4-209\,x^3+121\,x^2\right )\,{\ln \left (x+1\right )}^2+\left (1440\,x^3+384\,x^2-1056\,x\right )\,\ln \left (x+1\right )+2304\,x+2304} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(576*x - log(x + 1)^2*(180*x^2 + 180*x^3) + log(x + 1)*(576*x + 576))/(2304*x + log(x + 1)^2*(121*x^2 - 2
09*x^3 - 105*x^4 + 225*x^5) + log(x + 1)*(384*x^2 - 1056*x + 1440*x^3) + 2304),x)

[Out]

int(-(576*x - log(x + 1)^2*(180*x^2 + 180*x^3) + log(x + 1)*(576*x + 576))/(2304*x + log(x + 1)^2*(121*x^2 - 2
09*x^3 - 105*x^4 + 225*x^5) + log(x + 1)*(384*x^2 - 1056*x + 1440*x^3) + 2304), x)

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sympy [A]  time = 0.30, size = 31, normalized size = 1.03 \begin {gather*} \frac {576}{720 x + \left (225 x^{3} - 330 x^{2} + 121 x\right ) \log {\left (x + 1 \right )} - 528} - \frac {180}{225 x - 165} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((180*x**3+180*x**2)*ln(x+1)**2+(-576*x-576)*ln(x+1)-576*x)/((225*x**5-105*x**4-209*x**3+121*x**2)*l
n(x+1)**2+(1440*x**3+384*x**2-1056*x)*ln(x+1)+2304*x+2304),x)

[Out]

576/(720*x + (225*x**3 - 330*x**2 + 121*x)*log(x + 1) - 528) - 180/(225*x - 165)

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