3.95.71 \(\int \frac {14 x+36 x^2+28 x^3+6 x^4+(16+7 x+17 x^2+7 x^3+x^4) \log (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8})}{64+28 x+68 x^2+28 x^3+4 x^4+(-64-28 x-68 x^2-28 x^3-4 x^4) \log (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8})+(16+7 x+17 x^2+7 x^3+x^4) \log ^2(\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8})} \, dx\)

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

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Rubi [F]  time = 1.50, 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 {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(14*x + 36*x^2 + 28*x^3 + 6*x^4 + (16 + 7*x + 17*x^2 + 7*x^3 + x^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 +
238*x^5 + 82*x^6 + 14*x^7 + x^8)])/(64 + 28*x + 68*x^2 + 28*x^3 + 4*x^4 + (-64 - 28*x - 68*x^2 - 28*x^3 - 4*x
^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)] + (16 + 7*x + 17*x^2 + 7*x^3 + x^
4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)]^2),x]

[Out]

8*Defer[Int][(2 + Log[1/(x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])^(-2), x] - ((128*I)*Defer[Int][1/((-7 + I*Sqrt[15
] - 2*x)*(2 + Log[1/(x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])^2), x])/Sqrt[15] - I*Defer[Int][1/((I - x)*(2 + Log[1
/(x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])^2), x] - I*Defer[Int][1/((I + x)*(2 + Log[1/(x^2*(1 + x^2)*(16 + 7*x + x
^2)^2)])^2), x] - (14*(15 + (7*I)*Sqrt[15])*Defer[Int][1/((7 - I*Sqrt[15] + 2*x)*(2 + Log[1/(x^2*(1 + x^2)*(16
+ 7*x + x^2)^2)])^2), x])/15 - ((128*I)*Defer[Int][1/((7 + I*Sqrt[15] + 2*x)*(2 + Log[1/(x^2*(1 + x^2)*(16 +
7*x + x^2)^2)])^2), x])/Sqrt[15] - (14*(15 - (7*I)*Sqrt[15])*Defer[Int][1/((7 + I*Sqrt[15] + 2*x)*(2 + Log[1/(
x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])^2), x])/15 + Defer[Int][(2 + Log[1/(x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])^(-
1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (32+21 x+52 x^2+28 x^3+5 x^4\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )}{\left (16+7 x+17 x^2+7 x^3+x^4\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\\ &=\int \left (\frac {2 \left (16+14 x+35 x^2+21 x^3+4 x^4\right )}{\left (1+x^2\right ) \left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )}\right ) \, dx\\ &=2 \int \frac {16+14 x+35 x^2+21 x^3+4 x^4}{\left (1+x^2\right ) \left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ &=2 \int \left (\frac {4}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}-\frac {1}{\left (1+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {-32-7 x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ &=-\left (2 \int \frac {1}{\left (1+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )+2 \int \frac {-32-7 x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ &=-\left (2 \int \left (\frac {i}{2 (i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {i}{2 (i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx\right )+2 \int \left (-\frac {32}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}-\frac {7 x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )-i \int \frac {1}{(i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-14 \int \frac {x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-64 \int \frac {1}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )-i \int \frac {1}{(i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-14 \int \left (\frac {1+\frac {7 i}{\sqrt {15}}}{\left (7-i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {1-\frac {7 i}{\sqrt {15}}}{\left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx-64 \int \left (\frac {2 i}{\sqrt {15} \left (-7+i \sqrt {15}-2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {2 i}{\sqrt {15} \left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )-i \int \frac {1}{(i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-\frac {(128 i) \int \frac {1}{\left (-7+i \sqrt {15}-2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx}{\sqrt {15}}-\frac {(128 i) \int \frac {1}{\left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx}{\sqrt {15}}-\frac {1}{15} \left (14 \left (15-7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-\frac {1}{15} \left (14 \left (15+7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (7-i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 28, normalized size = 0.88 \begin {gather*} \frac {x}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(14*x + 36*x^2 + 28*x^3 + 6*x^4 + (16 + 7*x + 17*x^2 + 7*x^3 + x^4)*Log[E^4/(256*x^2 + 224*x^3 + 337
*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)])/(64 + 28*x + 68*x^2 + 28*x^3 + 4*x^4 + (-64 - 28*x - 68*x^2 - 28*x^3
- 4*x^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)] + (16 + 7*x + 17*x^2 + 7*x^
3 + x^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)]^2),x]

[Out]

x/(2 + Log[1/(x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])

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fricas [A]  time = 0.49, size = 46, normalized size = 1.44 \begin {gather*} \frac {x}{\log \left (\frac {e^{4}}{x^{8} + 14 \, x^{7} + 82 \, x^{6} + 238 \, x^{5} + 337 \, x^{4} + 224 \, x^{3} + 256 \, x^{2}}\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+6*x^4+28*
x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+
(-4*x^4-28*x^3-68*x^2-28*x-64)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+4*x^4+28*x^3+68
*x^2+28*x+64),x, algorithm="fricas")

[Out]

x/(log(e^4/(x^8 + 14*x^7 + 82*x^6 + 238*x^5 + 337*x^4 + 224*x^3 + 256*x^2)) - 2)

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giac [A]  time = 0.42, size = 42, normalized size = 1.31 \begin {gather*} -\frac {x}{\log \left (x^{8} + 14 \, x^{7} + 82 \, x^{6} + 238 \, x^{5} + 337 \, x^{4} + 224 \, x^{3} + 256 \, x^{2}\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+6*x^4+28*
x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+
(-4*x^4-28*x^3-68*x^2-28*x-64)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+4*x^4+28*x^3+68
*x^2+28*x+64),x, algorithm="giac")

[Out]

-x/(log(x^8 + 14*x^7 + 82*x^6 + 238*x^5 + 337*x^4 + 224*x^3 + 256*x^2) - 2)

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maple [A]  time = 0.13, size = 47, normalized size = 1.47

method	result	size
norman	\(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{8}+14 x^{7}+82 x^{6}+238 x^{5}+337 x^{4}+224 x^{3}+256 x^{2}}\right )-2}\)	\(47\)
risch	\(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{8}+14 x^{7}+82 x^{6}+238 x^{5}+337 x^{4}+224 x^{3}+256 x^{2}}\right )-2}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4+7*x^3+17*x^2+7*x+16)*ln(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+6*x^4+28*x^3+36*
x^2+14*x)/((x^4+7*x^3+17*x^2+7*x+16)*ln(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+(-4*x^4-
28*x^3-68*x^2-28*x-64)*ln(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+4*x^4+28*x^3+68*x^2+28*x
+64),x,method=_RETURNVERBOSE)

[Out]

x/(ln(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))-2)

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maxima [A]  time = 0.49, size = 28, normalized size = 0.88 \begin {gather*} -\frac {x}{2 \, \log \left (x^{2} + 7 \, x + 16\right ) + \log \left (x^{2} + 1\right ) + 2 \, \log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+6*x^4+28*
x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+
(-4*x^4-28*x^3-68*x^2-28*x-64)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))+4*x^4+28*x^3+68
*x^2+28*x+64),x, algorithm="maxima")

[Out]

-x/(2*log(x^2 + 7*x + 16) + log(x^2 + 1) + 2*log(x) - 2)

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mupad [B]  time = 8.67, size = 164, normalized size = 5.12 \begin {gather*} \frac {7\,\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )}{32\,\left (\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2\right )}-\frac {7}{16\,\left (\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2\right )}-\frac {x}{\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((14*x + 36*x^2 + 28*x^3 + 6*x^4 + log(exp(4)/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^
8))*(7*x + 17*x^2 + 7*x^3 + x^4 + 16))/(28*x - log(exp(4)/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14
*x^7 + x^8))*(28*x + 68*x^2 + 28*x^3 + 4*x^4 + 64) + log(exp(4)/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^
6 + 14*x^7 + x^8))^2*(7*x + 17*x^2 + 7*x^3 + x^4 + 16) + 68*x^2 + 28*x^3 + 4*x^4 + 64),x)

[Out]

(7*log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8))/(32*(log(256*x^2 + 224*x^3 + 337*x^4 +
238*x^5 + 82*x^6 + 14*x^7 + x^8) - 2)) - 7/(16*(log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 +
x^8) - 2)) - x/(log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8) - 2)

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sympy [A]  time = 0.33, size = 41, normalized size = 1.28 \begin {gather*} \frac {x}{\log {\left (\frac {e^{4}}{x^{8} + 14 x^{7} + 82 x^{6} + 238 x^{5} + 337 x^{4} + 224 x^{3} + 256 x^{2}} \right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4+7*x**3+17*x**2+7*x+16)*ln(exp(4)/(x**8+14*x**7+82*x**6+238*x**5+337*x**4+224*x**3+256*x**2))+
6*x**4+28*x**3+36*x**2+14*x)/((x**4+7*x**3+17*x**2+7*x+16)*ln(exp(4)/(x**8+14*x**7+82*x**6+238*x**5+337*x**4+2
24*x**3+256*x**2))**2+(-4*x**4-28*x**3-68*x**2-28*x-64)*ln(exp(4)/(x**8+14*x**7+82*x**6+238*x**5+337*x**4+224*
x**3+256*x**2))+4*x**4+28*x**3+68*x**2+28*x+64),x)

[Out]

x/(log(exp(4)/(x**8 + 14*x**7 + 82*x**6 + 238*x**5 + 337*x**4 + 224*x**3 + 256*x**2)) - 2)

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