3.28.22 \(\int \frac {-6 x^2+6 x^3+8 x^4+(-12 x-18 x^2-6 x^3) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+(-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8) \log (1+x)+(576 x^4+960 x^5+448 x^6+64 x^7) \log ^2(1+x)} \, dx\) [2722]

Optimal. Leaf size=26 \[ \frac {4}{16+\frac {2}{x^2 \left (x^2-(3+x) \log (1+x)\right )}} \]

[Out]

4/(16+2/(x^2-(3+x)*ln(1+x))/x^2)

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Rubi [F]
time = 2.70, 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 {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*x^2 + 6*x^3 + 8*x^4 + (-12*x - 18*x^2 - 6*x^3)*Log[1 + x])/(1 + x + 16*x^4 + 16*x^5 + 64*x^8 + 64*x^9
+ (-48*x^2 - 64*x^3 - 16*x^4 - 384*x^6 - 512*x^7 - 128*x^8)*Log[1 + x] + (576*x^4 + 960*x^5 + 448*x^6 + 64*x^7
)*Log[1 + x]^2),x]

[Out]

58*Defer[Int][(1 + 8*x^4 - 24*x^2*Log[1 + x] - 8*x^3*Log[1 + x])^(-2), x] - Defer[Int][1/(x*(1 + 8*x^4 - 24*x^
2*Log[1 + x] - 8*x^3*Log[1 + x])^2), x]/2 - 22*Defer[Int][x/(1 + 8*x^4 - 24*x^2*Log[1 + x] - 8*x^3*Log[1 + x])
^2, x] + 4*Defer[Int][x^2/(1 + 8*x^4 - 24*x^2*Log[1 + x] - 8*x^3*Log[1 + x])^2, x] + 2*Defer[Int][x^3/(1 + 8*x
^4 - 24*x^2*Log[1 + x] - 8*x^3*Log[1 + x])^2, x] - 4*Defer[Int][1/((1 + x)*(1 + 8*x^4 - 24*x^2*Log[1 + x] - 8*
x^3*Log[1 + x])^2), x] - (649*Defer[Int][1/((3 + x)*(1 + 8*x^4 - 24*x^2*Log[1 + x] - 8*x^3*Log[1 + x])^2), x])
/4 + Defer[Int][1/((3 + x)*(1 + 8*x^4 - 24*x^2*Log[1 + x] - 8*x^3*Log[1 + x])), x]/4 + Defer[Int][(x + 8*x^5 -
8*x^3*(3 + x)*Log[1 + x])^(-1), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (x \left (-3+3 x+4 x^2\right )-3 \left (2+3 x+x^2\right ) \log (1+x)\right )}{(1+x) \left (1+8 x^4-8 x^2 (3+x) \log (1+x)\right )^2} \, dx\\ &=2 \int \frac {x \left (x \left (-3+3 x+4 x^2\right )-3 \left (2+3 x+x^2\right ) \log (1+x)\right )}{(1+x) \left (1+8 x^4-8 x^2 (3+x) \log (1+x)\right )^2} \, dx\\ &=2 \int \left (\frac {-6-9 x-3 x^2-72 x^3+48 x^5+8 x^6}{8 x (1+x) (3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}+\frac {3 (2+x)}{8 x (3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-6-9 x-3 x^2-72 x^3+48 x^5+8 x^6}{x (1+x) (3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+\frac {3}{4} \int \frac {2+x}{x (3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )} \, dx\\ &=\frac {1}{4} \int \left (\frac {232}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}-\frac {2}{x \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}-\frac {88 x}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}+\frac {16 x^2}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}+\frac {8 x^3}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}-\frac {16}{(1+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}-\frac {649}{(3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2}\right ) \, dx+\frac {3}{4} \int \left (\frac {2}{3 x \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )}+\frac {1}{3 (3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{(3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )} \, dx-\frac {1}{2} \int \frac {1}{x \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+\frac {1}{2} \int \frac {1}{x \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )} \, dx+2 \int \frac {x^3}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+4 \int \frac {x^2}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx-4 \int \frac {1}{(1+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx-22 \int \frac {x}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+58 \int \frac {1}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx-\frac {649}{4} \int \frac {1}{(3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {1}{(3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )} \, dx-\frac {1}{2} \int \frac {1}{x \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+\frac {1}{2} \int \frac {1}{x+8 x^5-8 x^3 (3+x) \log (1+x)} \, dx+2 \int \frac {x^3}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+4 \int \frac {x^2}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx-4 \int \frac {1}{(1+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx-22 \int \frac {x}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx+58 \int \frac {1}{\left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx-\frac {649}{4} \int \frac {1}{(3+x) \left (1+8 x^4-24 x^2 \log (1+x)-8 x^3 \log (1+x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.10, size = 25, normalized size = 0.96 \begin {gather*} -\frac {1}{4 \left (1+8 x^4-8 x^2 (3+x) \log (1+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*x^2 + 6*x^3 + 8*x^4 + (-12*x - 18*x^2 - 6*x^3)*Log[1 + x])/(1 + x + 16*x^4 + 16*x^5 + 64*x^8 + 6
4*x^9 + (-48*x^2 - 64*x^3 - 16*x^4 - 384*x^6 - 512*x^7 - 128*x^8)*Log[1 + x] + (576*x^4 + 960*x^5 + 448*x^6 +
64*x^7)*Log[1 + x]^2),x]

[Out]

-1/4*1/(1 + 8*x^4 - 8*x^2*(3 + x)*Log[1 + x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(26)=52\).
time = 0.96, size = 57, normalized size = 2.19

method	result	size
risch	\(-\frac {1}{4 \left (8 x^{4}-8 \ln \left (x +1\right ) x^{3}-24 \ln \left (x +1\right ) x^{2}+1\right )}\)	\(30\)
derivativedivides	\(\frac {1}{32 \ln \left (x +1\right ) \left (x +1\right )^{3}-32 \left (x +1\right )^{4}+128 \left (x +1\right )^{3}-96 \left (x +1\right ) \ln \left (x +1\right )-192 \left (x +1\right )^{2}+64 \ln \left (x +1\right )+128 x +92}\)	\(57\)
default	\(\frac {1}{32 \ln \left (x +1\right ) \left (x +1\right )^{3}-32 \left (x +1\right )^{4}+128 \left (x +1\right )^{3}-96 \left (x +1\right ) \ln \left (x +1\right )-192 \left (x +1\right )^{2}+64 \ln \left (x +1\right )+128 x +92}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^3-18*x^2-12*x)*ln(x+1)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x^6+960*x^5+576*x^4)*ln(x+1)^2+(-128*x^8-512
*x^7-384*x^6-16*x^4-64*x^3-48*x^2)*ln(x+1)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x,method=_RETURNVERBOSE)

[Out]

1/4/(8*ln(x+1)*(x+1)^3-8*(x+1)^4+32*(x+1)^3-24*(x+1)*ln(x+1)-48*(x+1)^2+16*ln(x+1)+32*x+23)

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Maxima [A]
time = 0.31, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \, {\left (8 \, x^{4} - 8 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x + 1\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-18*x^2-12*x)*log(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x^6+960*x^5+576*x^4)*log(1+x)^2+(-128
*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48*x^2)*log(1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x, algorithm="maxima")

[Out]

-1/4/(8*x^4 - 8*(x^3 + 3*x^2)*log(x + 1) + 1)

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Fricas [A]
time = 0.36, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \, {\left (8 \, x^{4} - 8 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x + 1\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-18*x^2-12*x)*log(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x^6+960*x^5+576*x^4)*log(1+x)^2+(-128
*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48*x^2)*log(1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x, algorithm="fricas")

[Out]

-1/4/(8*x^4 - 8*(x^3 + 3*x^2)*log(x + 1) + 1)

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Sympy [A]
time = 0.14, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{- 32 x^{4} + \left (32 x^{3} + 96 x^{2}\right ) \log {\left (x + 1 \right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**3-18*x**2-12*x)*ln(1+x)+8*x**4+6*x**3-6*x**2)/((64*x**7+448*x**6+960*x**5+576*x**4)*ln(1+x)*
*2+(-128*x**8-512*x**7-384*x**6-16*x**4-64*x**3-48*x**2)*ln(1+x)+64*x**9+64*x**8+16*x**5+16*x**4+x+1),x)

[Out]

1/(-32*x**4 + (32*x**3 + 96*x**2)*log(x + 1) - 4)

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Giac [A]
time = 0.40, size = 29, normalized size = 1.12 \begin {gather*} -\frac {1}{4 \, {\left (8 \, x^{4} - 8 \, x^{3} \log \left (x + 1\right ) - 24 \, x^{2} \log \left (x + 1\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-18*x^2-12*x)*log(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x^6+960*x^5+576*x^4)*log(1+x)^2+(-128
*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48*x^2)*log(1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x, algorithm="giac")

[Out]

-1/4/(8*x^4 - 8*x^3*log(x + 1) - 24*x^2*log(x + 1) + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (x+1\right )\,\left (6\,x^3+18\,x^2+12\,x\right )+6\,x^2-6\,x^3-8\,x^4}{x+{\ln \left (x+1\right )}^2\,\left (64\,x^7+448\,x^6+960\,x^5+576\,x^4\right )-\ln \left (x+1\right )\,\left (128\,x^8+512\,x^7+384\,x^6+16\,x^4+64\,x^3+48\,x^2\right )+16\,x^4+16\,x^5+64\,x^8+64\,x^9+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + 1)*(12*x + 18*x^2 + 6*x^3) + 6*x^2 - 6*x^3 - 8*x^4)/(x + log(x + 1)^2*(576*x^4 + 960*x^5 + 448*x
^6 + 64*x^7) - log(x + 1)*(48*x^2 + 64*x^3 + 16*x^4 + 384*x^6 + 512*x^7 + 128*x^8) + 16*x^4 + 16*x^5 + 64*x^8
+ 64*x^9 + 1),x)

[Out]

int(-(log(x + 1)*(12*x + 18*x^2 + 6*x^3) + 6*x^2 - 6*x^3 - 8*x^4)/(x + log(x + 1)^2*(576*x^4 + 960*x^5 + 448*x
^6 + 64*x^7) - log(x + 1)*(48*x^2 + 64*x^3 + 16*x^4 + 384*x^6 + 512*x^7 + 128*x^8) + 16*x^4 + 16*x^5 + 64*x^8
+ 64*x^9 + 1), x)

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