Optimal. Leaf size=21 \[ x^2+\frac {\left (5+\frac {\log (-5 x)}{e^3}\right )^2}{\log (x)} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.19, 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 {-25 e^6-10 e^3 \log (-5 x)-\log ^2(-5 x)+\left (10 e^3+2 \log (-5 x)\right ) \log (x)+2 e^6 x^2 \log ^2(x)}{e^6 x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-25 e^6-10 e^3 \log (-5 x)-\log ^2(-5 x)+\left (10 e^3+2 \log (-5 x)\right ) \log (x)+2 e^6 x^2 \log ^2(x)}{x \log ^2(x)} \, dx}{e^6}\\ &=\frac {\int \left (2 e^6 x-\frac {\left (5 e^3+\log (-5 x)\right )^2}{x \log ^2(x)}+\frac {2 \left (5 e^3+\log (-5 x)\right )}{x \log (x)}\right ) \, dx}{e^6}\\ &=x^2-\frac {\int \frac {\left (5 e^3+\log (-5 x)\right )^2}{x \log ^2(x)} \, dx}{e^6}+\frac {2 \int \frac {5 e^3+\log (-5 x)}{x \log (x)} \, dx}{e^6}\\ &=x^2+\frac {2 \left (5 e^3+\log (-5 x)\right ) \log (\log (x))}{e^6}-\frac {\int \frac {\left (5 e^3+\log (-5 x)\right )^2}{x \log ^2(x)} \, dx}{e^6}-\frac {2 \int \frac {\log (\log (x))}{x} \, dx}{e^6}\\ &=x^2+\frac {2 \log (x)}{e^6}+\frac {2 \left (5 e^3+\log (-5 x)\right ) \log (\log (x))}{e^6}-\frac {2 \log (x) \log (\log (x))}{e^6}-\frac {\int \frac {\left (5 e^3+\log (-5 x)\right )^2}{x \log ^2(x)} \, dx}{e^6}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 35, normalized size = 1.67 \begin {gather*} \frac {e^6 x^2+\frac {\left (5 e^3+\log (-5 x)-\log (x)\right )^2}{\log (x)}+\log (x)}{e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs.
\(2(22)=44\).
time = 0.65, size = 79, normalized size = 3.76
method | result | size |
default | \({\mathrm e}^{-6} \left (\frac {2 \ln \left (5\right ) \ln \left (-x \right )}{\ln \left (x \right )}+\frac {10 \ln \left (5\right ) {\mathrm e}^{3}}{\ln \left (x \right )}+\frac {\ln \left (-x \right )^{2}}{\ln \left (x \right )}+\frac {25 \,{\mathrm e}^{6}}{\ln \left (x \right )}+\frac {10 \,{\mathrm e}^{3} \ln \left (-x \right )}{\ln \left (x \right )}+x^{2} {\mathrm e}^{6}+\frac {\ln \left (5\right )^{2}}{\ln \left (x \right )}\right )\) | \(79\) |
risch | \(x^{2}+{\mathrm e}^{-6} \ln \left (x \right )+\frac {{\mathrm e}^{-6} \left (-4 \pi ^{2}+40 \,{\mathrm e}^{3} \ln \left (5\right )+40 i {\mathrm e}^{3} \pi \mathrm {csgn}\left (i x \right )^{3}-40 i {\mathrm e}^{3} \pi \mathrm {csgn}\left (i x \right )^{2}+40 i {\mathrm e}^{3} \pi -8 i \ln \left (5\right ) \pi \mathrm {csgn}\left (i x \right )^{2}+8 i \ln \left (5\right ) \pi +8 i \ln \left (5\right ) \pi \mathrm {csgn}\left (i x \right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{6}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{4}+8 \pi ^{2} \mathrm {csgn}\left (i x \right )^{5}-8 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3}+100 \,{\mathrm e}^{6}+4 \ln \left (5\right )^{2}+8 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2}\right )}{4 \ln \left (x \right )}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (20) = 40\).
time = 0.28, size = 41, normalized size = 1.95 \begin {gather*} {\left (x^{2} e^{6} + \frac {10 \, e^{3} \log \left (-5 \, x\right )}{\log \left (x\right )} + \frac {\log \left (-5 \, x\right )^{2}}{\log \left (x\right )} + \frac {25 \, e^{6}}{\log \left (x\right )}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (20) = 40\).
time = 0.37, size = 143, normalized size = 6.81 \begin {gather*} \frac {\pi ^{2} x^{2} e^{6} - 10 \, \pi ^{2} e^{3} + {\left (x^{2} e^{6} - 10 \, e^{3}\right )} \log \left (5\right )^{2} - \log \left (5\right )^{3} + {\left (x^{2} e^{6} - 2 \, \log \left (5\right )\right )} \log \left (-5 \, x\right )^{2} + \log \left (-5 \, x\right )^{3} - {\left (\pi ^{2} + 25 \, e^{6}\right )} \log \left (5\right ) - {\left (2 \, {\left (x^{2} e^{6} - 5 \, e^{3}\right )} \log \left (5\right ) - 2 \, \log \left (5\right )^{2} - 25 \, e^{6}\right )} \log \left (-5 \, x\right )}{\pi ^{2} e^{6} + e^{6} \log \left (5\right )^{2} - 2 \, e^{6} \log \left (5\right ) \log \left (-5 \, x\right ) + e^{6} \log \left (-5 \, x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.07, size = 58, normalized size = 2.76 \begin {gather*} \frac {x^{2} e^{6} + \log {\left (x \right )}}{e^{6}} + \frac {- \pi ^{2} + \log {\left (5 \right )}^{2} + 10 e^{3} \log {\left (5 \right )} + 25 e^{6} + 2 i \pi \log {\left (5 \right )} + 10 i \pi e^{3}}{e^{6} \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs.
\(2 (20) = 40\).
time = 0.39, size = 139, normalized size = 6.62 \begin {gather*} \frac {{\left (\pi ^{2} x^{2} e^{6} \mathrm {sgn}\left (x\right ) - \pi ^{2} x^{2} e^{6} - 2 \, x^{2} e^{6} \log \left ({\left | x \right |}\right )^{2} + 20 \, \pi ^{2} e^{3} \mathrm {sgn}\left (x\right ) + 4 \, \pi ^{2} \log \left (5\right ) \mathrm {sgn}\left (x\right ) + \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - 20 \, \pi ^{2} e^{3} - 4 \, \pi ^{2} \log \left (5\right ) + 7 \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 20 \, e^{3} \log \left (5\right ) \log \left ({\left | x \right |}\right ) - 2 \, \log \left (5\right )^{2} \log \left ({\left | x \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right )^{3} - 50 \, e^{6} \log \left ({\left | x \right |}\right )\right )} e^{\left (-6\right )}}{\pi ^{2} \mathrm {sgn}\left (x\right ) - \pi ^{2} - 2 \, \log \left ({\left | x \right |}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.23, size = 49, normalized size = 2.33 \begin {gather*} 2\,{\mathrm {e}}^{-6}\,\ln \left (x\right )-{\mathrm {e}}^{-6}\,\left (2\,\ln \left (x\right )-x^2\,{\mathrm {e}}^6\right )+\frac {{\mathrm {e}}^{-6}\,\left ({\ln \left (-5\,x\right )}^2+10\,{\mathrm {e}}^3\,\ln \left (-5\,x\right )+25\,{\mathrm {e}}^6\right )}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________