3.50.12 \(\int \frac {e^3 (-5-x)+e^3 (-5-x) \log (x)+5 e^3 \log (x) \log (x \log (x))+e^3 (4-10 x-x^2) \log (x) \log ^2(x \log (x))}{x^2 \log (x)+(-8 x-2 x^3) \log (x) \log (x \log (x))+(16+8 x^2+x^4) \log (x) \log ^2(x \log (x))} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^3 (5+x)}{4+x^2-\frac {x}{\log (x \log (x))}} \]

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Rubi [F]  time = 13.40, 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 {e^3 (-5-x)+e^3 (-5-x) \log (x)+5 e^3 \log (x) \log (x \log (x))+e^3 \left (4-10 x-x^2\right ) \log (x) \log ^2(x \log (x))}{x^2 \log (x)+\left (-8 x-2 x^3\right ) \log (x) \log (x \log (x))+\left (16+8 x^2+x^4\right ) \log (x) \log ^2(x \log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^3*(-5 - x) + E^3*(-5 - x)*Log[x] + 5*E^3*Log[x]*Log[x*Log[x]] + E^3*(4 - 10*x - x^2)*Log[x]*Log[x*Log[x
]]^2)/(x^2*Log[x] + (-8*x - 2*x^3)*Log[x]*Log[x*Log[x]] + (16 + 8*x^2 + x^4)*Log[x]*Log[x*Log[x]]^2),x]

[Out]

(E^3*(5 + x))/(4 + x^2) + (1 - (15*I)/4)*E^3*Defer[Int][1/((2*I - x)*(-x + 4*Log[x*Log[x]] + x^2*Log[x*Log[x]]
)), x] - (1 + (15*I)/4)*E^3*Defer[Int][1/((2*I + x)*(-x + 4*Log[x*Log[x]] + x^2*Log[x*Log[x]])), x] + 80*E^3*D
efer[Int][1/((4 + x^2)^2*(-x + 4*Log[x*Log[x]] + x^2*Log[x*Log[x]])), x] + 16*E^3*Defer[Int][x/((4 + x^2)^2*(-
x + 4*Log[x*Log[x]] + x^2*Log[x*Log[x]])), x] - 6*E^3*Defer[Int][(x - (4 + x^2)*Log[x*Log[x]])^(-2), x] + (5/2
+ 3*I)*E^3*Defer[Int][1/((2*I - x)*(x - (4 + x^2)*Log[x*Log[x]])^2), x] - E^3*Defer[Int][x/(x - (4 + x^2)*Log
[x*Log[x]])^2, x] - (5/2 - 3*I)*E^3*Defer[Int][1/((2*I + x)*(x - (4 + x^2)*Log[x*Log[x]])^2), x] - 32*E^3*Defe
r[Int][1/((4 + x^2)^2*(x - (4 + x^2)*Log[x*Log[x]])^2), x] + 40*E^3*Defer[Int][x/((4 + x^2)^2*(x - (4 + x^2)*L
og[x*Log[x]])^2), x] - 5*E^3*Defer[Int][1/(Log[x]*(x - (4 + x^2)*Log[x*Log[x]])^2), x] - E^3*Defer[Int][x/(Log
[x]*(x - (4 + x^2)*Log[x*Log[x]])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^3 \left (-5-x-\log (x) \left (5+x-5 \log (x \log (x))+\left (-4+10 x+x^2\right ) \log ^2(x \log (x))\right )\right )}{\log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx\\ &=e^3 \int \frac {-5-x-\log (x) \left (5+x-5 \log (x \log (x))+\left (-4+10 x+x^2\right ) \log ^2(x \log (x))\right )}{\log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx\\ &=e^3 \int \left (\frac {4-10 x-x^2}{\left (4+x^2\right )^2}-\frac {(5+x) \left (16+8 x^2+x^4+16 \log (x)-4 x \log (x)+8 x^2 \log (x)+x^3 \log (x)+x^4 \log (x)\right )}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {20+8 x-15 x^2-2 x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx\\ &=e^3 \int \frac {4-10 x-x^2}{\left (4+x^2\right )^2} \, dx-e^3 \int \frac {(5+x) \left (16+8 x^2+x^4+16 \log (x)-4 x \log (x)+8 x^2 \log (x)+x^3 \log (x)+x^4 \log (x)\right )}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx+e^3 \int \frac {20+8 x-15 x^2-2 x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}-\frac {1}{8} e^3 \int 0 \, dx-e^3 \int \frac {(5+x) \left (\left (4+x^2\right )^2+\left (16-4 x+8 x^2+x^3+x^4\right ) \log (x)\right )}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx+e^3 \int \left (\frac {16 (5+x)}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}+\frac {-15-2 x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}+e^3 \int \frac {-15-2 x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx-e^3 \int \left (\frac {80}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}-\frac {4 x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {36 x^2}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {13 x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {6 x^4}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {x^5}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {80}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {16 x}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {40 x^2}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {8 x^3}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {5 x^4}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {x^5}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}\right ) \, dx+\left (16 e^3\right ) \int \frac {5+x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx+e^3 \int \left (-\frac {15}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}-\frac {2 x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx+\left (4 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (5 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (6 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (8 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (13 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (16 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx+\left (16 e^3\right ) \int \left (\frac {5}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}+\frac {x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx-\left (36 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (40 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (2 e^3\right ) \int \frac {x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx+\left (4 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (5 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (6 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (8 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (13 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (15 e^3\right ) \int \frac {1}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx+\left (16 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx-\left (16 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (36 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (40 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx+\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (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 = 0.33, size = 29, normalized size = 1.21 \begin {gather*} \frac {e^3 (5+x) \log (x \log (x))}{-x+\left (4+x^2\right ) \log (x \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3*(-5 - x) + E^3*(-5 - x)*Log[x] + 5*E^3*Log[x]*Log[x*Log[x]] + E^3*(4 - 10*x - x^2)*Log[x]*Log[x
*Log[x]]^2)/(x^2*Log[x] + (-8*x - 2*x^3)*Log[x]*Log[x*Log[x]] + (16 + 8*x^2 + x^4)*Log[x]*Log[x*Log[x]]^2),x]

[Out]

(E^3*(5 + x)*Log[x*Log[x]])/(-x + (4 + x^2)*Log[x*Log[x]])

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fricas [A]  time = 0.56, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (x + 5\right )} e^{3} \log \left (x \log \relax (x)\right )}{{\left (x^{2} + 4\right )} \log \left (x \log \relax (x)\right ) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-10*x+4)*exp(3)*log(x)*log(x*log(x))^2+5*exp(3)*log(x)*log(x*log(x))+(-x-5)*exp(3)*log(x)+(-x-
5)*exp(3))/((x^4+8*x^2+16)*log(x)*log(x*log(x))^2+(-2*x^3-8*x)*log(x)*log(x*log(x))+x^2*log(x)),x, algorithm="
fricas")

[Out]

(x + 5)*e^3*log(x*log(x))/((x^2 + 4)*log(x*log(x)) - x)

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giac [B]  time = 0.40, size = 56, normalized size = 2.33 \begin {gather*} \frac {x e^{3} \log \relax (x) + x e^{3} \log \left (\log \relax (x)\right ) + 5 \, e^{3} \log \relax (x) + 5 \, e^{3} \log \left (\log \relax (x)\right )}{x^{2} \log \relax (x) + x^{2} \log \left (\log \relax (x)\right ) - x + 4 \, \log \relax (x) + 4 \, \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-10*x+4)*exp(3)*log(x)*log(x*log(x))^2+5*exp(3)*log(x)*log(x*log(x))+(-x-5)*exp(3)*log(x)+(-x-
5)*exp(3))/((x^4+8*x^2+16)*log(x)*log(x*log(x))^2+(-2*x^3-8*x)*log(x)*log(x*log(x))+x^2*log(x)),x, algorithm="
giac")

[Out]

(x*e^3*log(x) + x*e^3*log(log(x)) + 5*e^3*log(x) + 5*e^3*log(log(x)))/(x^2*log(x) + x^2*log(log(x)) - x + 4*lo
g(x) + 4*log(log(x)))

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maple [C]  time = 0.24, size = 212, normalized size = 8.83

method	result	size
risch	\(\frac {{\mathrm e}^{3} \left (5+x \right )}{x^{2}+4}+\frac {2 i {\mathrm e}^{3} \left (5+x \right ) x}{\left (x^{2}+4\right ) \left (\pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )-\pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-\pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+\pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+4 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )-4 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-4 \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+4 \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+2 i x^{2} \ln \relax (x )+2 i x^{2} \ln \left (\ln \relax (x )\right )-2 i x +8 i \ln \relax (x )+8 i \ln \left (\ln \relax (x )\right )\right )}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-10*x+4)*exp(3)*ln(x)*ln(x*ln(x))^2+5*exp(3)*ln(x)*ln(x*ln(x))+(-x-5)*exp(3)*ln(x)+(-x-5)*exp(3))/((
x^4+8*x^2+16)*ln(x)*ln(x*ln(x))^2+(-2*x^3-8*x)*ln(x)*ln(x*ln(x))+x^2*ln(x)),x,method=_RETURNVERBOSE)

[Out]

exp(3)*(5+x)/(x^2+4)+2*I*exp(3)*(5+x)*x/(x^2+4)/(Pi*x^2*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))-Pi*x^2*csgn(I*
x)*csgn(I*x*ln(x))^2-Pi*x^2*csgn(I*ln(x))*csgn(I*x*ln(x))^2+Pi*x^2*csgn(I*x*ln(x))^3+4*Pi*csgn(I*x)*csgn(I*ln(
x))*csgn(I*x*ln(x))-4*Pi*csgn(I*x)*csgn(I*x*ln(x))^2-4*Pi*csgn(I*ln(x))*csgn(I*x*ln(x))^2+4*Pi*csgn(I*x*ln(x))
^3+2*I*x^2*ln(x)+2*I*x^2*ln(ln(x))-2*I*x+8*I*ln(x)+8*I*ln(ln(x)))

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maxima [B]  time = 0.41, size = 50, normalized size = 2.08 \begin {gather*} \frac {{\left (x e^{3} + 5 \, e^{3}\right )} \log \relax (x) + {\left (x e^{3} + 5 \, e^{3}\right )} \log \left (\log \relax (x)\right )}{{\left (x^{2} + 4\right )} \log \relax (x) + {\left (x^{2} + 4\right )} \log \left (\log \relax (x)\right ) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-10*x+4)*exp(3)*log(x)*log(x*log(x))^2+5*exp(3)*log(x)*log(x*log(x))+(-x-5)*exp(3)*log(x)+(-x-
5)*exp(3))/((x^4+8*x^2+16)*log(x)*log(x*log(x))^2+(-2*x^3-8*x)*log(x)*log(x*log(x))+x^2*log(x)),x, algorithm="
maxima")

[Out]

((x*e^3 + 5*e^3)*log(x) + (x*e^3 + 5*e^3)*log(log(x)))/((x^2 + 4)*log(x) + (x^2 + 4)*log(log(x)) - x)

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mupad [B]  time = 4.59, size = 33, normalized size = 1.38 \begin {gather*} \frac {\ln \left (x\,\ln \relax (x)\right )\,{\mathrm {e}}^3\,\left (x+5\right )}{4\,\ln \left (x\,\ln \relax (x)\right )-x+x^2\,\ln \left (x\,\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3)*(x + 5) + exp(3)*log(x)*(x + 5) - 5*log(x*log(x))*exp(3)*log(x) + log(x*log(x))^2*exp(3)*log(x)*(
10*x + x^2 - 4))/(x^2*log(x) + log(x*log(x))^2*log(x)*(8*x^2 + x^4 + 16) - log(x*log(x))*log(x)*(8*x + 2*x^3))
,x)

[Out]

(log(x*log(x))*exp(3)*(x + 5))/(4*log(x*log(x)) - x + x^2*log(x*log(x)))

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sympy [B]  time = 0.54, size = 53, normalized size = 2.21 \begin {gather*} \frac {x^{2} e^{3} + 5 x e^{3}}{- x^{3} - 4 x + \left (x^{4} + 8 x^{2} + 16\right ) \log {\left (x \log {\relax (x )} \right )}} - \frac {- x e^{3} - 5 e^{3}}{x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-10*x+4)*exp(3)*ln(x)*ln(x*ln(x))**2+5*exp(3)*ln(x)*ln(x*ln(x))+(-x-5)*exp(3)*ln(x)+(-x-5)*ex
p(3))/((x**4+8*x**2+16)*ln(x)*ln(x*ln(x))**2+(-2*x**3-8*x)*ln(x)*ln(x*ln(x))+x**2*ln(x)),x)

[Out]

(x**2*exp(3) + 5*x*exp(3))/(-x**3 - 4*x + (x**4 + 8*x**2 + 16)*log(x*log(x))) - (-x*exp(3) - 5*exp(3))/(x**2 +
4)

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