Optimal. Leaf size=20 \[ x^2 \log \left (-x+\frac {9 (-5-x)}{\log (5)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps
used = 10, number of rules used = 5, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6, 6820, 14,
45, 2442} \begin {gather*} x^2 \log \left (-\frac {x (9+\log (5))}{\log (5)}-\frac {45}{\log (5)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 14
Rule 45
Rule 2442
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 x^2+x^2 \log (5)+\left (90 x+18 x^2+2 x^2 \log (5)\right ) \log \left (\frac {-45-9 x-x \log (5)}{\log (5)}\right )}{45+x (9+\log (5))} \, dx\\ &=\int \frac {x^2 (9+\log (5))+\left (90 x+18 x^2+2 x^2 \log (5)\right ) \log \left (\frac {-45-9 x-x \log (5)}{\log (5)}\right )}{45+x (9+\log (5))} \, dx\\ &=\int x \left (\frac {x (9+\log (5))}{45+x (9+\log (5))}+2 \log \left (-\frac {45+x (9+\log (5))}{\log (5)}\right )\right ) \, dx\\ &=\int \left (\frac {x^2 (9+\log (5))}{45+x (9+\log (5))}+2 x \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )\right ) \, dx\\ &=2 \int x \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right ) \, dx+(9+\log (5)) \int \frac {x^2}{45+x (9+\log (5))} \, dx\\ &=x^2 \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )+(9+\log (5)) \int \left (-\frac {45}{(9+\log (5))^2}+\frac {x}{9+\log (5)}+\frac {2025}{(9+\log (5))^2 (45+x (9+\log (5)))}\right ) \, dx+\frac {(9+\log (5)) \int \frac {x^2}{-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}} \, dx}{\log (5)}\\ &=\frac {x^2}{2}-\frac {45 x}{9+\log (5)}+\frac {2025 \log (45+x (9+\log (5)))}{(9+\log (5))^2}+x^2 \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )+\frac {(9+\log (5)) \int \left (\frac {45 \log (5)}{(9+\log (5))^2}-\frac {x \log (5)}{9+\log (5)}+\frac {2025 \log (5)}{(9+\log (5))^2 (-45-x (9+\log (5)))}\right ) \, dx}{\log (5)}\\ &=x^2 \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} x^2 \log \left (-\frac {45+x (9+\log (5))}{\log (5)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs.
\(2(20)=40\).
time = 1.12, size = 236, normalized size = 11.80
method | result | size |
norman | \(x^{2} \ln \left (\frac {-x \ln \left (5\right )-9 x -45}{\ln \left (5\right )}\right )\) | \(21\) |
risch | \(x^{2} \ln \left (\frac {-x \ln \left (5\right )-9 x -45}{\ln \left (5\right )}\right )\) | \(21\) |
derivativedivides | \(-\frac {\ln \left (5\right ) \left (-\frac {2 \ln \left (5\right ) \left (\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2} \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{2}-\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{4}\right )}{\ln \left (5\right )+9}-\frac {90 \left (\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right ) \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )+\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}+\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {\ln \left (5\right ) \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{2 \left (\ln \left (5\right )+9\right )}-\frac {90 \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {2025 \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right ) \left (\ln \left (5\right )+9\right )}\right )}{\ln \left (5\right )+9}\) | \(236\) |
default | \(-\frac {\ln \left (5\right ) \left (-\frac {2 \ln \left (5\right ) \left (\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2} \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{2}-\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{4}\right )}{\ln \left (5\right )+9}-\frac {90 \left (\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right ) \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )+\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}+\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {\ln \left (5\right ) \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{2 \left (\ln \left (5\right )+9\right )}-\frac {90 \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {2025 \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right ) \left (\ln \left (5\right )+9\right )}\right )}{\ln \left (5\right )+9}\) | \(236\) |
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. 577 vs.
\(2 (18) = 36\).
time = 0.47, size = 577, normalized size = 28.85 \begin {gather*} {\left (\frac {x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81} + \frac {4050 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729}\right )} \log \left (5\right ) \log \left (-x - \frac {9 \, x}{\log \left (5\right )} - \frac {45}{\log \left (5\right )}\right ) + \frac {1}{2} \, {\left (\frac {x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81} + \frac {4050 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729}\right )} \log \left (5\right ) - \frac {{\left ({\left (\log \left (5\right )^{3} + 18 \, \log \left (5\right )^{2} + 81 \, \log \left (5\right )\right )} x^{2} + 4050 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )^{2} - 270 \, {\left (\log \left (5\right )^{2} + 9 \, \log \left (5\right )\right )} x + 12150 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )\right )} {\left (\frac {9}{\log \left (5\right )} + 1\right )} \log \left (5\right )}{2 \, {\left (\log \left (5\right )^{4} + 36 \, \log \left (5\right )^{3} + 486 \, \log \left (5\right )^{2} + 2916 \, \log \left (5\right ) + 6561\right )}} + 9 \, {\left (\frac {x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81} + \frac {4050 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729}\right )} \log \left (-x - \frac {9 \, x}{\log \left (5\right )} - \frac {45}{\log \left (5\right )}\right ) + 90 \, {\left (\frac {x}{\log \left (5\right ) + 9} - \frac {45 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81}\right )} \log \left (-x - \frac {9 \, x}{\log \left (5\right )} - \frac {45}{\log \left (5\right )}\right ) - \frac {9 \, {\left ({\left (\log \left (5\right )^{3} + 18 \, \log \left (5\right )^{2} + 81 \, \log \left (5\right )\right )} x^{2} + 4050 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )^{2} - 270 \, {\left (\log \left (5\right )^{2} + 9 \, \log \left (5\right )\right )} x + 12150 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )\right )} {\left (\frac {9}{\log \left (5\right )} + 1\right )}}{2 \, {\left (\log \left (5\right )^{4} + 36 \, \log \left (5\right )^{3} + 486 \, \log \left (5\right )^{2} + 2916 \, \log \left (5\right ) + 6561\right )}} + \frac {45 \, {\left (45 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )^{2} - 2 \, {\left (\log \left (5\right )^{2} + 9 \, \log \left (5\right )\right )} x + 90 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )\right )} {\left (\frac {9}{\log \left (5\right )} + 1\right )}}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729} + \frac {9 \, {\left (x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x\right )}}{2 \, {\left (\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81\right )}} + \frac {18225 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 20, normalized size = 1.00 \begin {gather*} x^{2} \log \left (-\frac {x \log \left (5\right ) + 9 \, x + 45}{\log \left (5\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 19, normalized size = 0.95 \begin {gather*} x^{2} \log {\left (\frac {- 9 x - x \log {\left (5 \right )} - 45}{\log {\left (5 \right )}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 24, normalized size = 1.20 \begin {gather*} x^{2} \log \left (-x \log \left (5\right ) - 9 \, x - 45\right ) - x^{2} \log \left (\log \left (5\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.23, size = 22, normalized size = 1.10 \begin {gather*} -x^2\,\left (\ln \left (\ln \left (5\right )\right )-\ln \left (-9\,x-x\,\ln \left (5\right )-45\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________