3.94.3 \(\int \frac {44 x+72 x^2+48 x^3+e^{4 x} (-19 x-30 x^2-24 x^3)+e^{8 x} (2 x+3 x^2+3 x^3)+(-19 x-30 x^2-24 x^3+e^{4 x} (4 x+6 x^2+6 x^3)) \log (x)+(2 x+3 x^2+3 x^3) \log ^2(x)+(2+3 x+3 x^2+e^{4 x} (8 x+12 x^2+12 x^3)) \log (2+3 x+3 x^2)}{32 x+48 x^2+48 x^3+e^{4 x} (-16 x-24 x^2-24 x^3)+e^{8 x} (2 x+3 x^2+3 x^3)+(-16 x-24 x^2-24 x^3+e^{4 x} (4 x+6 x^2+6 x^3)) \log (x)+(2 x+3 x^2+3 x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=28 \[ x+\frac {\log \left (2+3 \left (x+x^2\right )\right )}{4-e^{4 x}-\log (x)} \]

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Rubi [F]  time = 13.29, 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 {44 x+72 x^2+48 x^3+e^{4 x} \left (-19 x-30 x^2-24 x^3\right )+e^{8 x} \left (2 x+3 x^2+3 x^3\right )+\left (-19 x-30 x^2-24 x^3+e^{4 x} \left (4 x+6 x^2+6 x^3\right )\right ) \log (x)+\left (2 x+3 x^2+3 x^3\right ) \log ^2(x)+\left (2+3 x+3 x^2+e^{4 x} \left (8 x+12 x^2+12 x^3\right )\right ) \log \left (2+3 x+3 x^2\right )}{32 x+48 x^2+48 x^3+e^{4 x} \left (-16 x-24 x^2-24 x^3\right )+e^{8 x} \left (2 x+3 x^2+3 x^3\right )+\left (-16 x-24 x^2-24 x^3+e^{4 x} \left (4 x+6 x^2+6 x^3\right )\right ) \log (x)+\left (2 x+3 x^2+3 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(44*x + 72*x^2 + 48*x^3 + E^(4*x)*(-19*x - 30*x^2 - 24*x^3) + E^(8*x)*(2*x + 3*x^2 + 3*x^3) + (-19*x - 30*
x^2 - 24*x^3 + E^(4*x)*(4*x + 6*x^2 + 6*x^3))*Log[x] + (2*x + 3*x^2 + 3*x^3)*Log[x]^2 + (2 + 3*x + 3*x^2 + E^(
4*x)*(8*x + 12*x^2 + 12*x^3))*Log[2 + 3*x + 3*x^2])/(32*x + 48*x^2 + 48*x^3 + E^(4*x)*(-16*x - 24*x^2 - 24*x^3
) + E^(8*x)*(2*x + 3*x^2 + 3*x^3) + (-16*x - 24*x^2 - 24*x^3 + E^(4*x)*(4*x + 6*x^2 + 6*x^3))*Log[x] + (2*x +
3*x^2 + 3*x^3)*Log[x]^2),x]

[Out]

x - (6*I)*Sqrt[3/5]*Defer[Int][1/((-3 + I*Sqrt[15] - 6*x)*(-4 + E^(4*x) + Log[x])), x] - (6*(5 + I*Sqrt[15])*D
efer[Int][1/((3 - I*Sqrt[15] + 6*x)*(-4 + E^(4*x) + Log[x])), x])/5 - (6*I)*Sqrt[3/5]*Defer[Int][1/((3 + I*Sqr
t[15] + 6*x)*(-4 + E^(4*x) + Log[x])), x] - (6*(5 - I*Sqrt[15])*Defer[Int][1/((3 + I*Sqrt[15] + 6*x)*(-4 + E^(
4*x) + Log[x])), x])/5 + 16*Defer[Int][Log[2 + 3*x + 3*x^2]/(-4 + E^(4*x) + Log[x])^2, x] + Defer[Int][Log[2 +
3*x + 3*x^2]/(x*(-4 + E^(4*x) + Log[x])^2), x] - 4*Defer[Int][(Log[x]*Log[2 + 3*x + 3*x^2])/(-4 + E^(4*x) + L
og[x])^2, x] + 4*Defer[Int][Log[2 + 3*x + 3*x^2]/(-4 + E^(4*x) + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-4+e^{4 x}\right ) x \left (-11-18 x-12 x^2+e^{4 x} \left (2+3 x+3 x^2\right )\right )+x \left (-19-30 x-24 x^2+e^{4 x} \left (4+6 x+6 x^2\right )\right ) \log (x)+x \left (2+3 x+3 x^2\right ) \log ^2(x)+\left (1+4 e^{4 x} x\right ) \left (2+3 x+3 x^2\right ) \log \left (2+3 x+3 x^2\right )}{x \left (2+3 x+3 x^2\right ) \left (4-e^{4 x}-\log (x)\right )^2} \, dx\\ &=\int \left (1-\frac {(-1-16 x+4 x \log (x)) \log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2}+\frac {-3-6 x+8 \log \left (2+3 x+3 x^2\right )+12 x \log \left (2+3 x+3 x^2\right )+12 x^2 \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx\\ &=x-\int \frac {(-1-16 x+4 x \log (x)) \log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx+\int \frac {-3-6 x+8 \log \left (2+3 x+3 x^2\right )+12 x \log \left (2+3 x+3 x^2\right )+12 x^2 \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx\\ &=x+\int \frac {3+6 x-4 \left (2+3 x+3 x^2\right ) \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (4-e^{4 x}-\log (x)\right )} \, dx-\int \left (-\frac {16 \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2}-\frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2}+\frac {4 \log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2}\right ) \, dx\\ &=x-4 \int \frac {\log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+16 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+\int \frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx+\int \left (-\frac {3}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}-\frac {6 x}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {8 \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {12 x \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {12 x^2 \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx\\ &=x-3 \int \frac {1}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-4 \int \frac {\log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx-6 \int \frac {x}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+8 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+12 \int \frac {x \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+12 \int \frac {x^2 \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+16 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+\int \frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx\\ &=x-3 \int \left (\frac {2 i \sqrt {\frac {3}{5}}}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {2 i \sqrt {\frac {3}{5}}}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx-4 \int \frac {\log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx-6 \int \left (\frac {1+i \sqrt {\frac {3}{5}}}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {1-i \sqrt {\frac {3}{5}}}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx+8 \int \left (\frac {2 i \sqrt {\frac {3}{5}} \log \left (2+3 x+3 x^2\right )}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {2 i \sqrt {\frac {3}{5}} \log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx+12 \int \left (\frac {\left (1+i \sqrt {\frac {3}{5}}\right ) \log \left (2+3 x+3 x^2\right )}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {\left (1-i \sqrt {\frac {3}{5}}\right ) \log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx+12 \int \left (\frac {\log \left (2+3 x+3 x^2\right )}{3 \left (-4+e^{4 x}+\log (x)\right )}-\frac {(2+3 x) \log \left (2+3 x+3 x^2\right )}{3 \left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx+16 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+\int \frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx\\ &=x-4 \int \frac {\log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+4 \int \frac {\log \left (2+3 x+3 x^2\right )}{-4+e^{4 x}+\log (x)} \, dx-4 \int \frac {(2+3 x) \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+16 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx-\left (6 i \sqrt {\frac {3}{5}}\right ) \int \frac {1}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\left (6 i \sqrt {\frac {3}{5}}\right ) \int \frac {1}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\left (16 i \sqrt {\frac {3}{5}}\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\left (16 i \sqrt {\frac {3}{5}}\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\frac {1}{5} \left (6 \left (5-i \sqrt {15}\right )\right ) \int \frac {1}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\frac {1}{5} \left (12 \left (5-i \sqrt {15}\right )\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\frac {1}{5} \left (6 \left (5+i \sqrt {15}\right )\right ) \int \frac {1}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\frac {1}{5} \left (12 \left (5+i \sqrt {15}\right )\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\int \frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx\\ &=x-4 \int \frac {\log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+4 \int \frac {\log \left (2+3 x+3 x^2\right )}{-4+e^{4 x}+\log (x)} \, dx-4 \int \left (\frac {2 \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}+\frac {3 x \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )}\right ) \, dx+16 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx-\left (6 i \sqrt {\frac {3}{5}}\right ) \int \frac {1}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\left (6 i \sqrt {\frac {3}{5}}\right ) \int \frac {1}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\left (16 i \sqrt {\frac {3}{5}}\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\left (16 i \sqrt {\frac {3}{5}}\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\frac {1}{5} \left (6 \left (5-i \sqrt {15}\right )\right ) \int \frac {1}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\frac {1}{5} \left (12 \left (5-i \sqrt {15}\right )\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\frac {1}{5} \left (6 \left (5+i \sqrt {15}\right )\right ) \int \frac {1}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\frac {1}{5} \left (12 \left (5+i \sqrt {15}\right )\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\int \frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx\\ &=x-4 \int \frac {\log (x) \log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx+4 \int \frac {\log \left (2+3 x+3 x^2\right )}{-4+e^{4 x}+\log (x)} \, dx-8 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-12 \int \frac {x \log \left (2+3 x+3 x^2\right )}{\left (2+3 x+3 x^2\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+16 \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-4+e^{4 x}+\log (x)\right )^2} \, dx-\left (6 i \sqrt {\frac {3}{5}}\right ) \int \frac {1}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\left (6 i \sqrt {\frac {3}{5}}\right ) \int \frac {1}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\left (16 i \sqrt {\frac {3}{5}}\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (-3+i \sqrt {15}-6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\left (16 i \sqrt {\frac {3}{5}}\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\frac {1}{5} \left (6 \left (5-i \sqrt {15}\right )\right ) \int \frac {1}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\frac {1}{5} \left (12 \left (5-i \sqrt {15}\right )\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3+i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx-\frac {1}{5} \left (6 \left (5+i \sqrt {15}\right )\right ) \int \frac {1}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\frac {1}{5} \left (12 \left (5+i \sqrt {15}\right )\right ) \int \frac {\log \left (2+3 x+3 x^2\right )}{\left (3-i \sqrt {15}+6 x\right ) \left (-4+e^{4 x}+\log (x)\right )} \, dx+\int \frac {\log \left (2+3 x+3 x^2\right )}{x \left (-4+e^{4 x}+\log (x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 26, normalized size = 0.93 \begin {gather*} x-\frac {\log \left (2+3 x+3 x^2\right )}{-4+e^{4 x}+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(44*x + 72*x^2 + 48*x^3 + E^(4*x)*(-19*x - 30*x^2 - 24*x^3) + E^(8*x)*(2*x + 3*x^2 + 3*x^3) + (-19*x
- 30*x^2 - 24*x^3 + E^(4*x)*(4*x + 6*x^2 + 6*x^3))*Log[x] + (2*x + 3*x^2 + 3*x^3)*Log[x]^2 + (2 + 3*x + 3*x^2
+ E^(4*x)*(8*x + 12*x^2 + 12*x^3))*Log[2 + 3*x + 3*x^2])/(32*x + 48*x^2 + 48*x^3 + E^(4*x)*(-16*x - 24*x^2 -
24*x^3) + E^(8*x)*(2*x + 3*x^2 + 3*x^3) + (-16*x - 24*x^2 - 24*x^3 + E^(4*x)*(4*x + 6*x^2 + 6*x^3))*Log[x] + (
2*x + 3*x^2 + 3*x^3)*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.84, size = 38, normalized size = 1.36 \begin {gather*} \frac {x e^{\left (4 \, x\right )} + x \log \relax (x) - 4 \, x - \log \left (3 \, x^{2} + 3 \, x + 2\right )}{e^{\left (4 \, x\right )} + \log \relax (x) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x^3+12*x^2+8*x)*exp(2*x)^2+3*x^2+3*x+2)*log(3*x^2+3*x+2)+(3*x^3+3*x^2+2*x)*log(x)^2+((6*x^3+6*
x^2+4*x)*exp(2*x)^2-24*x^3-30*x^2-19*x)*log(x)+(3*x^3+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-30*x^2-19*x)*exp(2*x)^2+4
8*x^3+72*x^2+44*x)/((3*x^3+3*x^2+2*x)*log(x)^2+((6*x^3+6*x^2+4*x)*exp(2*x)^2-24*x^3-24*x^2-16*x)*log(x)+(3*x^3
+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-24*x^2-16*x)*exp(2*x)^2+48*x^3+48*x^2+32*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 38, normalized size = 1.36 \begin {gather*} \frac {x e^{\left (4 \, x\right )} + x \log \relax (x) - 4 \, x - \log \left (3 \, x^{2} + 3 \, x + 2\right )}{e^{\left (4 \, x\right )} + \log \relax (x) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x^3+12*x^2+8*x)*exp(2*x)^2+3*x^2+3*x+2)*log(3*x^2+3*x+2)+(3*x^3+3*x^2+2*x)*log(x)^2+((6*x^3+6*
x^2+4*x)*exp(2*x)^2-24*x^3-30*x^2-19*x)*log(x)+(3*x^3+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-30*x^2-19*x)*exp(2*x)^2+4
8*x^3+72*x^2+44*x)/((3*x^3+3*x^2+2*x)*log(x)^2+((6*x^3+6*x^2+4*x)*exp(2*x)^2-24*x^3-24*x^2-16*x)*log(x)+(3*x^3
+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-24*x^2-16*x)*exp(2*x)^2+48*x^3+48*x^2+32*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 26, normalized size = 0.93

method	result	size
risch	\(-\frac {\ln \left (3 x^{2}+3 x +2\right )}{{\mathrm e}^{4 x}+\ln \relax (x )-4}+x\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((12*x^3+12*x^2+8*x)*exp(2*x)^2+3*x^2+3*x+2)*ln(3*x^2+3*x+2)+(3*x^3+3*x^2+2*x)*ln(x)^2+((6*x^3+6*x^2+4*x)
*exp(2*x)^2-24*x^3-30*x^2-19*x)*ln(x)+(3*x^3+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-30*x^2-19*x)*exp(2*x)^2+48*x^3+72*
x^2+44*x)/((3*x^3+3*x^2+2*x)*ln(x)^2+((6*x^3+6*x^2+4*x)*exp(2*x)^2-24*x^3-24*x^2-16*x)*ln(x)+(3*x^3+3*x^2+2*x)
*exp(2*x)^4+(-24*x^3-24*x^2-16*x)*exp(2*x)^2+48*x^3+48*x^2+32*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.60, size = 38, normalized size = 1.36 \begin {gather*} \frac {x e^{\left (4 \, x\right )} + x \log \relax (x) - 4 \, x - \log \left (3 \, x^{2} + 3 \, x + 2\right )}{e^{\left (4 \, x\right )} + \log \relax (x) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x^3+12*x^2+8*x)*exp(2*x)^2+3*x^2+3*x+2)*log(3*x^2+3*x+2)+(3*x^3+3*x^2+2*x)*log(x)^2+((6*x^3+6*
x^2+4*x)*exp(2*x)^2-24*x^3-30*x^2-19*x)*log(x)+(3*x^3+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-30*x^2-19*x)*exp(2*x)^2+4
8*x^3+72*x^2+44*x)/((3*x^3+3*x^2+2*x)*log(x)^2+((6*x^3+6*x^2+4*x)*exp(2*x)^2-24*x^3-24*x^2-16*x)*log(x)+(3*x^3
+3*x^2+2*x)*exp(2*x)^4+(-24*x^3-24*x^2-16*x)*exp(2*x)^2+48*x^3+48*x^2+32*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.05, size = 25, normalized size = 0.89 \begin {gather*} x-\frac {\ln \left (3\,x^2+3\,x+2\right )}{{\mathrm {e}}^{4\,x}+\ln \relax (x)-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((44*x - log(x)*(19*x - exp(4*x)*(4*x + 6*x^2 + 6*x^3) + 30*x^2 + 24*x^3) + exp(8*x)*(2*x + 3*x^2 + 3*x^3)
- exp(4*x)*(19*x + 30*x^2 + 24*x^3) + log(x)^2*(2*x + 3*x^2 + 3*x^3) + 72*x^2 + 48*x^3 + log(3*x + 3*x^2 + 2)*
(3*x + exp(4*x)*(8*x + 12*x^2 + 12*x^3) + 3*x^2 + 2))/(32*x - log(x)*(16*x - exp(4*x)*(4*x + 6*x^2 + 6*x^3) +
24*x^2 + 24*x^3) + exp(8*x)*(2*x + 3*x^2 + 3*x^3) - exp(4*x)*(16*x + 24*x^2 + 24*x^3) + log(x)^2*(2*x + 3*x^2
+ 3*x^3) + 48*x^2 + 48*x^3),x)

[Out]

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

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sympy [A]  time = 0.54, size = 22, normalized size = 0.79 \begin {gather*} x - \frac {\log {\left (3 x^{2} + 3 x + 2 \right )}}{e^{4 x} + \log {\relax (x )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x**3+12*x**2+8*x)*exp(2*x)**2+3*x**2+3*x+2)*ln(3*x**2+3*x+2)+(3*x**3+3*x**2+2*x)*ln(x)**2+((6*
x**3+6*x**2+4*x)*exp(2*x)**2-24*x**3-30*x**2-19*x)*ln(x)+(3*x**3+3*x**2+2*x)*exp(2*x)**4+(-24*x**3-30*x**2-19*
x)*exp(2*x)**2+48*x**3+72*x**2+44*x)/((3*x**3+3*x**2+2*x)*ln(x)**2+((6*x**3+6*x**2+4*x)*exp(2*x)**2-24*x**3-24
*x**2-16*x)*ln(x)+(3*x**3+3*x**2+2*x)*exp(2*x)**4+(-24*x**3-24*x**2-16*x)*exp(2*x)**2+48*x**3+48*x**2+32*x),x)

[Out]

x - log(3*x**2 + 3*x + 2)/(exp(4*x) + log(x) - 4)

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