3.95.11 \(\int \frac {176+124 x+20 x^2+e^x (-88-62 x-10 x^2)+(88+256 x+134 x^2+20 x^3+e^x (-44-216 x-129 x^2-20 x^3)+e^{2 x} (44 x+31 x^2+5 x^3)) \log (x^2)+e^x (44 x+31 x^2+5 x^3) \log (x^2) \log (\frac {(22 x+10 x^2) \log (x^2)}{20+5 x})}{(176 x+124 x^2+20 x^3+e^x (-176 x-124 x^2-20 x^3)+e^{2 x} (44 x+31 x^2+5 x^3)) \log (x^2)} \, dx\)

Optimal. Leaf size=35 \[ x+\frac {\log \left (\left (x+\frac {x+\left (-\frac {3}{5}+x\right ) x}{4+x}\right ) \log \left (x^2\right )\right )}{2-e^x} \]

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Rubi [F]  time = 39.70, 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 {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(176 + 124*x + 20*x^2 + E^x*(-88 - 62*x - 10*x^2) + (88 + 256*x + 134*x^2 + 20*x^3 + E^x*(-44 - 216*x - 12
9*x^2 - 20*x^3) + E^(2*x)*(44*x + 31*x^2 + 5*x^3))*Log[x^2] + E^x*(44*x + 31*x^2 + 5*x^3)*Log[x^2]*Log[((22*x
+ 10*x^2)*Log[x^2])/(20 + 5*x)])/((176*x + 124*x^2 + 20*x^3 + E^x*(-176*x - 124*x^2 - 20*x^3) + E^(2*x)*(44*x
+ 31*x^2 + 5*x^3))*Log[x^2]),x]

[Out]

x - Defer[Int][1/((-2 + E^x)*x), x] + Defer[Int][1/((-2 + E^x)*(4 + x)), x] - 5*Defer[Int][1/((-2 + E^x)*(11 +
5*x)), x] - 2*Defer[Int][1/((-2 + E^x)*x*Log[x^2]), x] + 2*Defer[Int][Log[(2*x*(11 + 5*x)*Log[x^2])/(5*(4 + x
))]/(-2 + E^x)^2, x] + Defer[Int][Log[(2*x*(11 + 5*x)*Log[x^2])/(5*(4 + x))]/(-2 + E^x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {4-2 e^x}{\log \left (x^2\right )}+\frac {88+256 x+134 x^2+20 x^3+e^{2 x} x \left (44+31 x+5 x^2\right )-e^x \left (44+216 x+129 x^2+20 x^3\right )+e^x x \left (44+31 x+5 x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{44+31 x+5 x^2}}{\left (2-e^x\right )^2 x} \, dx\\ &=\int \left (1+\frac {2 \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right )^2}+\frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right ) x (4+x) (11+5 x) \log \left (x^2\right )}\right ) \, dx\\ &=x+2 \int \frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right )^2} \, dx+\int \frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right ) x (4+x) (11+5 x) \log \left (x^2\right )} \, dx\\ &=x+2 \int \frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right )^2} \, dx+\int \frac {2 \left (44+31 x+5 x^2\right )-\log \left (x^2\right ) \left (-44-40 x-5 x^2+x \left (44+31 x+5 x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )\right )}{\left (2-e^x\right ) x (4+x) (11+5 x) \log \left (x^2\right )} \, dx\\ &=x+2 \int \frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right )^2} \, dx+\int \left (\frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{44 \left (-2+e^x\right ) x \log \left (x^2\right )}+\frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{36 \left (-2+e^x\right ) (4+x) \log \left (x^2\right )}-\frac {25 \left (-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )\right )}{99 \left (-2+e^x\right ) (11+5 x) \log \left (x^2\right )}\right ) \, dx\\ &=x+\frac {1}{44} \int \frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right ) x \log \left (x^2\right )} \, dx+\frac {1}{36} \int \frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right ) (4+x) \log \left (x^2\right )} \, dx-\frac {25}{99} \int \frac {-88-62 x-10 x^2-44 \log \left (x^2\right )-40 x \log \left (x^2\right )-5 x^2 \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right ) (11+5 x) \log \left (x^2\right )} \, dx+2 \int \frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^x\right )^2} \, dx\\ &=x+\frac {1}{44} \int \frac {2 \left (44+31 x+5 x^2\right )-\log \left (x^2\right ) \left (-44-40 x-5 x^2+x \left (44+31 x+5 x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )\right )}{\left (2-e^x\right ) x \log \left (x^2\right )} \, dx+\frac {1}{36} \int \frac {2 \left (44+31 x+5 x^2\right )-\log \left (x^2\right ) \left (-44-40 x-5 x^2+x \left (44+31 x+5 x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )\right )}{\left (2-e^x\right ) (4+x) \log \left (x^2\right )} \, dx-\frac {25}{99} \int \frac {2 \left (44+31 x+5 x^2\right )-\log \left (x^2\right ) \left (-44-40 x-5 x^2+x \left (44+31 x+5 x^2\right ) \log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )\right )}{\left (2-e^x\right ) (11+5 x) \log \left (x^2\right )} \, dx+2 \int \frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{\left (-2+e^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 = 0.24, size = 31, normalized size = 0.89 \begin {gather*} x-\frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{-2+e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(176 + 124*x + 20*x^2 + E^x*(-88 - 62*x - 10*x^2) + (88 + 256*x + 134*x^2 + 20*x^3 + E^x*(-44 - 216*
x - 129*x^2 - 20*x^3) + E^(2*x)*(44*x + 31*x^2 + 5*x^3))*Log[x^2] + E^x*(44*x + 31*x^2 + 5*x^3)*Log[x^2]*Log[(
(22*x + 10*x^2)*Log[x^2])/(20 + 5*x)])/((176*x + 124*x^2 + 20*x^3 + E^x*(-176*x - 124*x^2 - 20*x^3) + E^(2*x)*
(44*x + 31*x^2 + 5*x^3))*Log[x^2]),x]

[Out]

x - Log[(2*x*(11 + 5*x)*Log[x^2])/(5*(4 + x))]/(-2 + E^x)

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fricas [A]  time = 0.61, size = 38, normalized size = 1.09 \begin {gather*} \frac {x e^{x} - 2 \, x - \log \left (\frac {2 \, {\left (5 \, x^{2} + 11 \, x\right )} \log \left (x^{2}\right )}{5 \, {\left (x + 4\right )}}\right )}{e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/(20+5*x))+((5*x^3+31*x^2+44*x)*exp(x
)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+20*x^3+134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+
176)/((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124*x^2+176*x)/log(x^2),x, algorithm=
"fricas")

[Out]

(x*e^x - 2*x - log(2/5*(5*x^2 + 11*x)*log(x^2)/(x + 4)))/(e^x - 2)

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giac [A]  time = 0.63, size = 42, normalized size = 1.20 \begin {gather*} \frac {x e^{x} - 2 \, x - \log \left (10 \, x \log \left (x^{2}\right ) + 22 \, \log \left (x^{2}\right )\right ) + \log \left (5 \, x + 20\right ) - \log \relax (x)}{e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/(20+5*x))+((5*x^3+31*x^2+44*x)*exp(x
)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+20*x^3+134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+
176)/((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124*x^2+176*x)/log(x^2),x, algorithm=
"giac")

[Out]

(x*e^x - 2*x - log(10*x*log(x^2) + 22*log(x^2)) + log(5*x + 20) - log(x))/(e^x - 2)

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maple [C]  time = 0.72, size = 1672, normalized size = 47.77

method	result	size
risch	\(-\frac {\ln \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{{\mathrm e}^{x}-2}+\frac {-i \pi \,\mathrm {csgn}\left (i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \left (x +\frac {11}{5}\right )}{4+x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \left (x +\frac {11}{5}\right )}{4+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{4+x}\right )-i \pi \mathrm {csgn}\left (\frac {\left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )^{2} \mathrm {csgn}\left (\frac {i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \left (x +\frac {11}{5}\right )}{4+x}\right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )^{2}+i \pi \mathrm {csgn}\left (i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{3}-i \pi \mathrm {csgn}\left (\frac {\left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (x +\frac {11}{5}\right )\right ) \mathrm {csgn}\left (i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {\left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \left (x +\frac {11}{5}\right )}{4+x}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {\left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )^{3}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \left (x +\frac {11}{5}\right )}{4+x}\right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x}{4+x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \left (x +\frac {11}{5}\right )}{4+x}\right ) \mathrm {csgn}\left (\frac {i}{4+x}\right )+i \pi \,\mathrm {csgn}\left (i \left (x +\frac {11}{5}\right )\right ) \mathrm {csgn}\left (i \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +\frac {11}{5}\right ) \left (4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )+i \pi +2 \,{\mathrm e}^{x} x -4 x -2 \ln \relax (x )+2 \ln \left (4+x \right )-2 \ln \left (x +\frac {11}{5}\right )}{2 \,{\mathrm e}^{x}-4}\)	\(1672\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^3+31*x^2+44*x)*exp(x)*ln(x^2)*ln((10*x^2+22*x)*ln(x^2)/(20+5*x))+((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*
x^3-129*x^2-216*x-44)*exp(x)+20*x^3+134*x^2+256*x+88)*ln(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+176)/((5*x
^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124*x^2+176*x)/ln(x^2),x,method=_RETURNVERBOSE)

[Out]

-1/(exp(x)-2)*ln(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)+1/2*(-I*P
i*csgn(I*(x+11/5)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I
/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(x+11/5))^2-I*Pi*c
sgn(I*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I*(x+11/5)*(4
*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2+I*Pi*csgn(I*(x+11/5)/(4+
x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*x)^3-I*Pi*csgn(I/(4+x)
*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(x+11/5))^2*csgn(I/(4+x)
)-I*Pi*csgn((x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2
)*x)^2*csgn(I*(x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)
^2)*x)-I*Pi*csgn(I/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*
(x+11/5))*csgn(I*(x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x
^2)^2)*x)^2+I*Pi*csgn(I*(x+11/5)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*
x^2)^2))^3-I*Pi*csgn((x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn
(I*x^2)^2)*x)^2-I*Pi*csgn(I*(x+11/5))*csgn(I*(x+11/5)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2
*Pi*csgn(I*x)*csgn(I*x^2)^2))^2+I*Pi*csgn((x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2
)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*x)*csgn(I*(x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2
)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*x)+I*Pi*csgn(I/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*
Pi*csgn(I*x)*csgn(I*x^2)^2)*(x+11/5))^3+I*Pi*csgn((x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*cs
gn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*x)^3+I*Pi*csgn(I*x)*csgn(I/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I
*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(x+11/5))*csgn(I*(x+11/5)/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi
*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*x)-I*Pi*csgn(I*x)*csgn(I*(x+11/5)/(4+x)*(4*I*ln(x)+Pi*c
sgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*x)^2+I*Pi*csgn(I*(x+11/5)*(4*I*ln(x)+Pi*
csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I/(4+x)*(4*I*ln(x)+Pi*csgn(I*x^2)
^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(x+11/5))*csgn(I/(4+x))+I*Pi*csgn(I*(x+11/5))*csgn
(I*(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I*(x+11/5)*(4*I*
ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))+I*Pi+2*exp(x)*x-4*x-2*ln(x)+2
*ln(4+x)-2*ln(x+11/5))/(exp(x)-2)

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maxima [A]  time = 0.51, size = 42, normalized size = 1.20 \begin {gather*} \frac {x e^{x} - 2 \, x + \log \relax (5) - 2 \, \log \relax (2) - \log \left (5 \, x + 11\right ) + \log \left (x + 4\right ) - \log \relax (x) - \log \left (\log \relax (x)\right )}{e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/(20+5*x))+((5*x^3+31*x^2+44*x)*exp(x
)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+20*x^3+134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+
176)/((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124*x^2+176*x)/log(x^2),x, algorithm=
"maxima")

[Out]

(x*e^x - 2*x + log(5) - 2*log(2) - log(5*x + 11) + log(x + 4) - log(x) - log(log(x)))/(e^x - 2)

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mupad [B]  time = 8.12, size = 32, normalized size = 0.91 \begin {gather*} x-\frac {\ln \left (\frac {\ln \left (x^2\right )\,\left (10\,x^2+22\,x\right )}{5\,x+20}\right )}{{\mathrm {e}}^x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((124*x + log(x^2)*(256*x + exp(2*x)*(44*x + 31*x^2 + 5*x^3) + 134*x^2 + 20*x^3 - exp(x)*(216*x + 129*x^2 +
20*x^3 + 44) + 88) - exp(x)*(62*x + 10*x^2 + 88) + 20*x^2 + log((log(x^2)*(22*x + 10*x^2))/(5*x + 20))*log(x^
2)*exp(x)*(44*x + 31*x^2 + 5*x^3) + 176)/(log(x^2)*(176*x + exp(2*x)*(44*x + 31*x^2 + 5*x^3) + 124*x^2 + 20*x^
3 - exp(x)*(176*x + 124*x^2 + 20*x^3))),x)

[Out]

x - log((log(x^2)*(22*x + 10*x^2))/(5*x + 20))/(exp(x) - 2)

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sympy [A]  time = 0.86, size = 26, normalized size = 0.74 \begin {gather*} x - \frac {\log {\left (\frac {\left (10 x^{2} + 22 x\right ) \log {\left (x^{2} \right )}}{5 x + 20} \right )}}{e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**3+31*x**2+44*x)*exp(x)*ln(x**2)*ln((10*x**2+22*x)*ln(x**2)/(20+5*x))+((5*x**3+31*x**2+44*x)*e
xp(x)**2+(-20*x**3-129*x**2-216*x-44)*exp(x)+20*x**3+134*x**2+256*x+88)*ln(x**2)+(-10*x**2-62*x-88)*exp(x)+20*
x**2+124*x+176)/((5*x**3+31*x**2+44*x)*exp(x)**2+(-20*x**3-124*x**2-176*x)*exp(x)+20*x**3+124*x**2+176*x)/ln(x
**2),x)

[Out]

x - log((10*x**2 + 22*x)*log(x**2)/(5*x + 20))/(exp(x) - 2)

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