3.16.7 \(\int \frac {60+49 x-4 x^2-3 x^3+(-60-17 x+4 x^2+x^3+e^x (-45 x-9 x^2+5 x^3+x^4)) \log (15 x+8 x^2+x^3) \log (\log (15 x+8 x^2+x^3))+(-15 x-8 x^2-x^3) \log (15 x+8 x^2+x^3) \log (\log (15 x+8 x^2+x^3)) \log (\frac {\log (\log (15 x+8 x^2+x^3))}{x})}{(15 x+8 x^2+x^3) \log (15 x+8 x^2+x^3) \log (\log (15 x+8 x^2+x^3))} \, dx\)

Optimal. Leaf size=27 \[ (4-x) \left (-e^x+\log \left (\frac {\log (\log (x (3+x) (5+x)))}{x}\right )\right ) \]

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Rubi [F]  time = 4.87, 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 {60+49 x-4 x^2-3 x^3+\left (-60-17 x+4 x^2+x^3+e^x \left (-45 x-9 x^2+5 x^3+x^4\right )\right ) \log \left (15 x+8 x^2+x^3\right ) \log \left (\log \left (15 x+8 x^2+x^3\right )\right )+\left (-15 x-8 x^2-x^3\right ) \log \left (15 x+8 x^2+x^3\right ) \log \left (\log \left (15 x+8 x^2+x^3\right )\right ) \log \left (\frac {\log \left (\log \left (15 x+8 x^2+x^3\right )\right )}{x}\right )}{\left (15 x+8 x^2+x^3\right ) \log \left (15 x+8 x^2+x^3\right ) \log \left (\log \left (15 x+8 x^2+x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(60 + 49*x - 4*x^2 - 3*x^3 + (-60 - 17*x + 4*x^2 + x^3 + E^x*(-45*x - 9*x^2 + 5*x^3 + x^4))*Log[15*x + 8*x
^2 + x^3]*Log[Log[15*x + 8*x^2 + x^3]] + (-15*x - 8*x^2 - x^3)*Log[15*x + 8*x^2 + x^3]*Log[Log[15*x + 8*x^2 +
x^3]]*Log[Log[Log[15*x + 8*x^2 + x^3]]/x])/((15*x + 8*x^2 + x^3)*Log[15*x + 8*x^2 + x^3]*Log[Log[15*x + 8*x^2
+ x^3]]),x]

[Out]

-E^x - E^x*(3 - x) - 4*Log[x] - x*Log[Log[Log[x*(15 + 8*x + x^2)]]/x] + 4*Defer[Int][1/(x*Log[x*(15 + 8*x + x^
2)]*Log[Log[x*(15 + 8*x + x^2)]]), x] + 4*Defer[Int][1/((3 + x)*Log[x*(15 + 8*x + x^2)]*Log[Log[x*(15 + 8*x +
x^2)]]), x] + 4*Defer[Int][1/((5 + x)*Log[x*(15 + 8*x + x^2)]*Log[Log[x*(15 + 8*x + x^2)]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {60+49 x-4 x^2-3 x^3+\left (-60-17 x+4 x^2+x^3+e^x \left (-45 x-9 x^2+5 x^3+x^4\right )\right ) \log \left (15 x+8 x^2+x^3\right ) \log \left (\log \left (15 x+8 x^2+x^3\right )\right )+\left (-15 x-8 x^2-x^3\right ) \log \left (15 x+8 x^2+x^3\right ) \log \left (\log \left (15 x+8 x^2+x^3\right )\right ) \log \left (\frac {\log \left (\log \left (15 x+8 x^2+x^3\right )\right )}{x}\right )}{x \left (15+8 x+x^2\right ) \log \left (15 x+8 x^2+x^3\right ) \log \left (\log \left (15 x+8 x^2+x^3\right )\right )} \, dx\\ &=\int \left (e^x (-3+x)-\frac {17}{15+8 x+x^2}-\frac {60}{x \left (15+8 x+x^2\right )}+\frac {4 x}{15+8 x+x^2}+\frac {x^2}{15+8 x+x^2}+\frac {49}{(3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}+\frac {60}{x (3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {4 x}{(3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {3 x^2}{(3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right )\right ) \, dx\\ &=-\left (3 \int \frac {x^2}{(3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx\right )+4 \int \frac {x}{15+8 x+x^2} \, dx-4 \int \frac {x}{(3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-17 \int \frac {1}{15+8 x+x^2} \, dx+49 \int \frac {1}{(3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-60 \int \frac {1}{x \left (15+8 x+x^2\right )} \, dx+60 \int \frac {1}{x (3+x) (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\int e^x (-3+x) \, dx+\int \frac {x^2}{15+8 x+x^2} \, dx-\int \log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right ) \, dx\\ &=-e^x (3-x)+x-x \log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right )-3 \int \left (\frac {1}{\log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}+\frac {9}{2 (3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {25}{2 (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}\right ) \, dx-4 \int \frac {1}{x} \, dx-4 \int \frac {-8-x}{15+8 x+x^2} \, dx-4 \int \left (-\frac {3}{2 (3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}+\frac {5}{2 (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}\right ) \, dx-6 \int \frac {1}{3+x} \, dx-\frac {17}{2} \int \frac {1}{3+x} \, dx+\frac {17}{2} \int \frac {1}{5+x} \, dx+10 \int \frac {1}{5+x} \, dx+49 \int \left (\frac {1}{2 (3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {1}{2 (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}\right ) \, dx+60 \int \left (\frac {1}{15 x \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {1}{6 (3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}+\frac {1}{10 (5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}\right ) \, dx-\int e^x \, dx+\int \frac {-15-8 x}{15+8 x+x^2} \, dx+\int \left (-1+\frac {15+16 x+3 x^2}{\left (15+8 x+x^2\right ) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}\right ) \, dx\\ &=-e^x-e^x (3-x)-4 \log (x)-\frac {29}{2} \log (3+x)+\frac {37}{2} \log (5+x)-x \log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right )-3 \int \frac {1}{\log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+4 \int \frac {1}{x \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {9}{2} \int \frac {1}{3+x} \, dx-6 \int \frac {1}{5+x} \, dx+6 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+6 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+10 \int \frac {1}{3+x} \, dx-10 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-10 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-\frac {25}{2} \int \frac {1}{5+x} \, dx-\frac {27}{2} \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {49}{2} \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-\frac {49}{2} \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {75}{2} \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\int \frac {15+16 x+3 x^2}{\left (15+8 x+x^2\right ) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx\\ &=-e^x-e^x (3-x)-4 \log (x)-x \log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right )-3 \int \frac {1}{\log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+4 \int \frac {1}{x \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+6 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+6 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-10 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-10 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-\frac {27}{2} \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {49}{2} \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-\frac {49}{2} \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {75}{2} \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\int \left (\frac {3}{\log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {3}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}-\frac {5}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}\right ) \, dx\\ &=-e^x-e^x (3-x)-4 \log (x)-x \log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right )-3 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+4 \int \frac {1}{x \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-5 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+6 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+6 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-10 \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-10 \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-\frac {27}{2} \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {49}{2} \int \frac {1}{(3+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx-\frac {49}{2} \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx+\frac {75}{2} \int \frac {1}{(5+x) \log \left (x \left (15+8 x+x^2\right )\right ) \log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 47, normalized size = 1.74 \begin {gather*} e^x (-4+x)-4 \log (x)+4 \log \left (\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )\right )-x \log \left (\frac {\log \left (\log \left (x \left (15+8 x+x^2\right )\right )\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(60 + 49*x - 4*x^2 - 3*x^3 + (-60 - 17*x + 4*x^2 + x^3 + E^x*(-45*x - 9*x^2 + 5*x^3 + x^4))*Log[15*x
+ 8*x^2 + x^3]*Log[Log[15*x + 8*x^2 + x^3]] + (-15*x - 8*x^2 - x^3)*Log[15*x + 8*x^2 + x^3]*Log[Log[15*x + 8*
x^2 + x^3]]*Log[Log[Log[15*x + 8*x^2 + x^3]]/x])/((15*x + 8*x^2 + x^3)*Log[15*x + 8*x^2 + x^3]*Log[Log[15*x +
8*x^2 + x^3]]),x]

[Out]

E^x*(-4 + x) - 4*Log[x] + 4*Log[Log[Log[x*(15 + 8*x + x^2)]]] - x*Log[Log[Log[x*(15 + 8*x + x^2)]]/x]

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fricas [A]  time = 0.74, size = 31, normalized size = 1.15 \begin {gather*} {\left (x - 4\right )} e^{x} - {\left (x - 4\right )} \log \left (\frac {\log \left (\log \left (x^{3} + 8 \, x^{2} + 15 \, x\right )\right )}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-8*x^2-15*x)*log(x^3+8*x^2+15*x)*log(log(x^3+8*x^2+15*x))*log(log(log(x^3+8*x^2+15*x))/x)+((x^
4+5*x^3-9*x^2-45*x)*exp(x)+x^3+4*x^2-17*x-60)*log(x^3+8*x^2+15*x)*log(log(x^3+8*x^2+15*x))-3*x^3-4*x^2+49*x+60
)/(x^3+8*x^2+15*x)/log(x^3+8*x^2+15*x)/log(log(x^3+8*x^2+15*x)),x, algorithm="fricas")

[Out]

(x - 4)*e^x - (x - 4)*log(log(log(x^3 + 8*x^2 + 15*x))/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{3} + 8 \, x^{2} + 15 \, x\right )} \log \left (x^{3} + 8 \, x^{2} + 15 \, x\right ) \log \left (\frac {\log \left (\log \left (x^{3} + 8 \, x^{2} + 15 \, x\right )\right )}{x}\right ) \log \left (\log \left (x^{3} + 8 \, x^{2} + 15 \, x\right )\right ) + 3 \, x^{3} - {\left (x^{3} + 4 \, x^{2} + {\left (x^{4} + 5 \, x^{3} - 9 \, x^{2} - 45 \, x\right )} e^{x} - 17 \, x - 60\right )} \log \left (x^{3} + 8 \, x^{2} + 15 \, x\right ) \log \left (\log \left (x^{3} + 8 \, x^{2} + 15 \, x\right )\right ) + 4 \, x^{2} - 49 \, x - 60}{{\left (x^{3} + 8 \, x^{2} + 15 \, x\right )} \log \left (x^{3} + 8 \, x^{2} + 15 \, x\right ) \log \left (\log \left (x^{3} + 8 \, x^{2} + 15 \, x\right )\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-8*x^2-15*x)*log(x^3+8*x^2+15*x)*log(log(x^3+8*x^2+15*x))*log(log(log(x^3+8*x^2+15*x))/x)+((x^
4+5*x^3-9*x^2-45*x)*exp(x)+x^3+4*x^2-17*x-60)*log(x^3+8*x^2+15*x)*log(log(x^3+8*x^2+15*x))-3*x^3-4*x^2+49*x+60
)/(x^3+8*x^2+15*x)/log(x^3+8*x^2+15*x)/log(log(x^3+8*x^2+15*x)),x, algorithm="giac")

[Out]

integrate(-((x^3 + 8*x^2 + 15*x)*log(x^3 + 8*x^2 + 15*x)*log(log(log(x^3 + 8*x^2 + 15*x))/x)*log(log(x^3 + 8*x
^2 + 15*x)) + 3*x^3 - (x^3 + 4*x^2 + (x^4 + 5*x^3 - 9*x^2 - 45*x)*e^x - 17*x - 60)*log(x^3 + 8*x^2 + 15*x)*log
(log(x^3 + 8*x^2 + 15*x)) + 4*x^2 - 49*x - 60)/((x^3 + 8*x^2 + 15*x)*log(x^3 + 8*x^2 + 15*x)*log(log(x^3 + 8*x
^2 + 15*x))), x)

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maple [C]  time = 0.78, size = 733, normalized size = 27.15

method	result	size
risch	\(-x \ln \left (\ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )\right )+x \ln \relax (x )+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )}{x}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )}{x}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )}{x}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )}{x}\right )^{3}}{2}-4 \ln \relax (x )+{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}+4 \ln \left (\ln \left (\ln \relax (x )+\ln \left (x^{2}+8 x +15\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x^{2}+8 x +15\right )\right )+\mathrm {csgn}\left (i \left (x^{2}+8 x +15\right )\right )\right )}{2}\right )\right )\)	\(733\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-8*x^2-15*x)*ln(x^3+8*x^2+15*x)*ln(ln(x^3+8*x^2+15*x))*ln(ln(ln(x^3+8*x^2+15*x))/x)+((x^4+5*x^3-9*x^
2-45*x)*exp(x)+x^3+4*x^2-17*x-60)*ln(x^3+8*x^2+15*x)*ln(ln(x^3+8*x^2+15*x))-3*x^3-4*x^2+49*x+60)/(x^3+8*x^2+15
*x)/ln(x^3+8*x^2+15*x)/ln(ln(x^3+8*x^2+15*x)),x,method=_RETURNVERBOSE)

[Out]

-x*ln(ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2+8*x+15))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(
x^2+8*x+15))+csgn(I*(x^2+8*x+15)))))+x*ln(x)+1/2*I*Pi*x*csgn(I/x)*csgn(I*ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn
(I*x*(x^2+8*x+15))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*(x^2+8*x+15)))))*csgn(I
/x*ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2+8*x+15))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x^2
+8*x+15))+csgn(I*(x^2+8*x+15)))))-1/2*I*Pi*x*csgn(I/x)*csgn(I/x*ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2
+8*x+15))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*(x^2+8*x+15)))))^2-1/2*I*Pi*x*cs
gn(I*ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2+8*x+15))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x
^2+8*x+15))+csgn(I*(x^2+8*x+15)))))*csgn(I/x*ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2+8*x+15))*(-csgn(I*
x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*(x^2+8*x+15)))))^2+1/2*I*Pi*x*csgn(I/x*ln(ln(x)+ln(
x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2+8*x+15))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x^2+8*x+15))+csgn(
I*(x^2+8*x+15)))))^3-4*ln(x)+exp(x)*x-4*exp(x)+4*ln(ln(ln(x)+ln(x^2+8*x+15)-1/2*I*Pi*csgn(I*x*(x^2+8*x+15))*(-
csgn(I*x*(x^2+8*x+15))+csgn(I*x))*(-csgn(I*x*(x^2+8*x+15))+csgn(I*(x^2+8*x+15)))))

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maxima [A]  time = 0.87, size = 31, normalized size = 1.15 \begin {gather*} {\left (x - 4\right )} e^{x} + {\left (x - 4\right )} \log \relax (x) - {\left (x - 4\right )} \log \left (\log \left (\log \left (x + 5\right ) + \log \left (x + 3\right ) + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-8*x^2-15*x)*log(x^3+8*x^2+15*x)*log(log(x^3+8*x^2+15*x))*log(log(log(x^3+8*x^2+15*x))/x)+((x^
4+5*x^3-9*x^2-45*x)*exp(x)+x^3+4*x^2-17*x-60)*log(x^3+8*x^2+15*x)*log(log(x^3+8*x^2+15*x))-3*x^3-4*x^2+49*x+60
)/(x^3+8*x^2+15*x)/log(x^3+8*x^2+15*x)/log(log(x^3+8*x^2+15*x)),x, algorithm="maxima")

[Out]

(x - 4)*e^x + (x - 4)*log(x) - (x - 4)*log(log(log(x + 5) + log(x + 3) + log(x)))

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mupad [B]  time = 1.65, size = 76, normalized size = 2.81 \begin {gather*} 4\,\ln \left (\ln \left (\ln \left (x^3+8\,x^2+15\,x\right )\right )\right )-4\,\ln \relax (x)+{\mathrm {e}}^x\,\left (x-4\right )-\frac {\ln \left (\frac {\ln \left (\ln \left (x^3+8\,x^2+15\,x\right )\right )}{x}\right )\,\left (x^4+8\,x^3+15\,x^2\right )}{x\,\left (x^2+8\,x+15\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x^2 - 49*x + 3*x^3 + log(log(15*x + 8*x^2 + x^3))*log(15*x + 8*x^2 + x^3)*(17*x + exp(x)*(45*x + 9*x^2
- 5*x^3 - x^4) - 4*x^2 - x^3 + 60) + log(log(15*x + 8*x^2 + x^3))*log(15*x + 8*x^2 + x^3)*log(log(log(15*x +
8*x^2 + x^3))/x)*(15*x + 8*x^2 + x^3) - 60)/(log(log(15*x + 8*x^2 + x^3))*log(15*x + 8*x^2 + x^3)*(15*x + 8*x^
2 + x^3)),x)

[Out]

4*log(log(log(15*x + 8*x^2 + x^3))) - 4*log(x) + exp(x)*(x - 4) - (log(log(log(15*x + 8*x^2 + x^3))/x)*(15*x^2
+ 8*x^3 + x^4))/(x*(8*x + x^2 + 15))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-8*x**2-15*x)*ln(x**3+8*x**2+15*x)*ln(ln(x**3+8*x**2+15*x))*ln(ln(ln(x**3+8*x**2+15*x))/x)+((
x**4+5*x**3-9*x**2-45*x)*exp(x)+x**3+4*x**2-17*x-60)*ln(x**3+8*x**2+15*x)*ln(ln(x**3+8*x**2+15*x))-3*x**3-4*x*
*2+49*x+60)/(x**3+8*x**2+15*x)/ln(x**3+8*x**2+15*x)/ln(ln(x**3+8*x**2+15*x)),x)

[Out]

Timed out

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