Optimal. Leaf size=29 \[ 9 \left (3+x-\frac {1}{\left (x+\frac {x}{x+5 (i \pi +\log (4))^2}\right )^2}\right ) \]
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Rubi [B] Both result and optimal contain complex but leaf count is larger than twice the leaf
count of optimal. \(182\) vs. \(2(29)=58\).
time = 0.33, antiderivative size = 182, normalized size of antiderivative = 6.28, number of steps
used = 3, number of rules used = 2, integrand size = 184, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6, 2099}
\begin {gather*} -\frac {225 (\pi -i \log (4))^4}{x^2 \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2}+9 x-\frac {90 (\pi -i \log (4))^2}{\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )}-\frac {9}{\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )^2}+\frac {90 (\pi -i \log (4))^2}{x \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+x^3 \left (1+125 (i \pi +\log (4))^6\right )} \, dx\\ &=\int \left (9+\frac {90 (\pi -i \log (4))^2}{x^2 \left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^3}+\frac {450 (\pi -i \log (4))^4}{x^3 \left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^2}-\frac {18}{\left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^2 \left (-1+5 \pi ^2-x-10 i \pi \log (4)-5 \log ^2(4)\right )^3}-\frac {90 (\pi -i \log (4))^2}{\left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^3 \left (-1+5 \pi ^2-x-10 i \pi \log (4)-5 \log ^2(4)\right )^2}\right ) \, dx\\ &=9 x+\frac {90 (\pi -i \log (4))^2}{x \left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^3}-\frac {225 (\pi -i \log (4))^4}{x^2 \left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^2}-\frac {9}{\left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^2 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )^2}-\frac {90 (\pi -i \log (4))^2}{\left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^3 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(163\) vs. \(2(29)=58\).
time = 0.14, size = 163, normalized size = 5.62 \begin {gather*} \frac {9 \left (-x^2+x^5+25 \pi ^4 \left (-1+x^3\right )-100 i \pi ^3 \left (-1+x^3\right ) \log (4)-10 x \log ^2(4)-25 \log ^4(4)+2 x^4 \left (1+5 \log ^2(4)\right )+x^3 \left (1+5 \log ^2(4)\right )^2+20 i \pi \log (4) \left (-x+x^4-5 \log ^2(4)+x^3 \left (1+5 \log ^2(4)\right )\right )-10 \pi ^2 \left (-x+x^4-15 \log ^2(4)+x^3 \left (1+15 \log ^2(4)\right )\right )\right )}{x^2 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 475 vs. \(2 (30 ) = 60\).
time = 2.44, size = 476, normalized size = 16.41 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 149 vs. \(2 (29) = 58\).
time = 0.31, size = 149, normalized size = 5.14 \begin {gather*} 9 \, x + \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \left (2\right ) - 600 \, \pi ^{2} \log \left (2\right )^{2} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 10 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (2\right ) - 4 \, \log \left (2\right )^{2}\right )} x + x^{2}\right )}}{2 \, {\left (5 \, \pi ^{2} - 20 i \, \pi \log \left (2\right ) - 20 \, \log \left (2\right )^{2} - 1\right )} x^{3} - x^{4} - {\left (25 \, \pi ^{4} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 40 \, {\left (15 \, \pi ^{2} - 1\right )} \log \left (2\right )^{2} - 10 \, \pi ^{2} - 40 \, {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} \log \left (2\right ) + 1\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 240 vs. \(2 (29) = 58\).
time = 0.36, size = 240, normalized size = 8.28 \begin {gather*} -\frac {9 \, {\left (2 \, {\left (5 \, \pi ^{2} - 1\right )} x^{4} - x^{5} - 400 \, {\left (x^{3} - 1\right )} \log \left (2\right )^{4} + 25 \, \pi ^{4} - {\left (25 \, \pi ^{4} - 10 \, \pi ^{2} + 1\right )} x^{3} + 800 \, {\left (i \, \pi - i \, \pi x^{3}\right )} \log \left (2\right )^{3} - 10 \, \pi ^{2} x + 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{3} - x^{4} - 15 \, \pi ^{2} + x\right )} \log \left (2\right )^{2} + x^{2} + 40 \, {\left (-i \, \pi x^{4} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{3} - 5 i \, \pi ^{3} + i \, \pi x\right )} \log \left (2\right )\right )}}{800 i \, \pi x^{2} \log \left (2\right )^{3} + 400 \, x^{2} \log \left (2\right )^{4} - 2 \, {\left (5 \, \pi ^{2} - 1\right )} x^{3} + x^{4} + {\left (25 \, \pi ^{4} - 10 \, \pi ^{2} + 1\right )} x^{2} - 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{2} - x^{3}\right )} \log \left (2\right )^{2} - 40 \, {\left (-i \, \pi x^{3} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{2}\right )} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 168 vs. \(2 (24) = 48\).
time = 18.39, size = 168, normalized size = 5.79 \begin {gather*} 9 x + \frac {- 9 x^{2} + x \left (- 360 \log {\left (2 \right )}^{2} + 90 \pi ^{2} - 360 i \pi \log {\left (2 \right )}\right ) - 225 \pi ^{4} - 3600 \log {\left (2 \right )}^{4} + 5400 \pi ^{2} \log {\left (2 \right )}^{2} - 7200 i \pi \log {\left (2 \right )}^{3} + 1800 i \pi ^{3} \log {\left (2 \right )}}{x^{4} + x^{3} \left (- 10 \pi ^{2} + 2 + 40 \log {\left (2 \right )}^{2} + 40 i \pi \log {\left (2 \right )}\right ) + x^{2} \left (- 600 \pi ^{2} \log {\left (2 \right )}^{2} - 10 \pi ^{2} + 1 + 40 \log {\left (2 \right )}^{2} + 400 \log {\left (2 \right )}^{4} + 25 \pi ^{4} - 200 i \pi ^{3} \log {\left (2 \right )} + 40 i \pi \log {\left (2 \right )} + 800 i \pi \log {\left (2 \right )}^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 93 vs. \(2 (29) = 58\).
time = 0.42, size = 93, normalized size = 3.21 \begin {gather*} 9 \, x - \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \left (2\right ) - 600 \, \pi ^{2} \log \left (2\right )^{2} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 10 \, \pi ^{2} x + 40 i \, \pi x \log \left (2\right ) + 40 \, x \log \left (2\right )^{2} + x^{2}\right )}}{{\left (5 \, \pi ^{2} x - 20 i \, \pi x \log \left (2\right ) - 20 \, x \log \left (2\right )^{2} - x^{2} - x\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 20.40, size = 640, normalized size = 22.07 \begin {gather*} 9\,x+\frac {900\,\Pi \,{\ln \left (4\right )}^3-900\,\Pi ^3\,\ln \left (4\right )-\Pi ^4\,225{}\mathrm {i}-{\ln \left (4\right )}^4\,225{}\mathrm {i}+\Pi ^2\,{\ln \left (4\right )}^2\,1350{}\mathrm {i}-\frac {90\,x\,\left (5\,\Pi ^4-\Pi ^2+{\ln \left (4\right )}^2-10\,{\ln \left (4\right )}^4-60\,\Pi ^2\,{\ln \left (2\right )}^2+45\,\Pi ^2\,{\ln \left (4\right )}^2+60\,{\ln \left (2\right )}^2\,{\ln \left (4\right )}^2-120\,\Pi ^2\,\ln \left (2\right )\,\ln \left (4\right )+\Pi \,\ln \left (4\right )\,2{}\mathrm {i}-\Pi ^3\,\ln \left (2\right )\,60{}\mathrm {i}-\Pi \,{\ln \left (4\right )}^3\,40{}\mathrm {i}+\Pi ^3\,\ln \left (4\right )\,10{}\mathrm {i}+\Pi \,\ln \left (2\right )\,{\ln \left (4\right )}^2\,60{}\mathrm {i}+\Pi \,{\ln \left (2\right )}^2\,\ln \left (4\right )\,120{}\mathrm {i}\right )}{\Pi ^2\,5{}\mathrm {i}+20\,\ln \left (2\right )\,\Pi -{\ln \left (2\right )}^2\,20{}\mathrm {i}-\mathrm {i}}+\frac {9\,x^2\,\left (25\,\Pi ^4-10\,\Pi ^2+40\,{\ln \left (2\right )}^2+400\,{\ln \left (2\right )}^4-450\,{\ln \left (4\right )}^4-2400\,\Pi ^2\,{\ln \left (2\right )}^2+2250\,\Pi ^2\,{\ln \left (4\right )}^2+1800\,{\ln \left (2\right )}^2\,{\ln \left (4\right )}^2-3600\,\Pi ^2\,\ln \left (2\right )\,\ln \left (4\right )+1+\Pi \,\ln \left (2\right )\,40{}\mathrm {i}+\Pi \,{\ln \left (2\right )}^3\,800{}\mathrm {i}-\Pi ^3\,\ln \left (2\right )\,2000{}\mathrm {i}-\Pi \,{\ln \left (4\right )}^3\,1800{}\mathrm {i}+\Pi ^3\,\ln \left (4\right )\,900{}\mathrm {i}+\Pi \,\ln \left (2\right )\,{\ln \left (4\right )}^2\,1800{}\mathrm {i}+\Pi \,{\ln \left (2\right )}^2\,\ln \left (4\right )\,3600{}\mathrm {i}\right )}{200\,\Pi ^3\,\ln \left (2\right )-800\,\Pi \,{\ln \left (2\right )}^3-40\,\Pi \,\ln \left (2\right )-\Pi ^2\,10{}\mathrm {i}+\Pi ^4\,25{}\mathrm {i}+{\ln \left (2\right )}^2\,40{}\mathrm {i}+{\ln \left (2\right )}^4\,400{}\mathrm {i}-\Pi ^2\,{\ln \left (2\right )}^2\,600{}\mathrm {i}+1{}\mathrm {i}}+\frac {2700\,x^3\,\left (2\,\Pi ^3\,\ln \left (4\right )-4\,\Pi \,{\ln \left (4\right )}^3-4\,\Pi ^3\,\ln \left (2\right )+{\ln \left (4\right )}^4\,1{}\mathrm {i}+\Pi ^2\,{\ln \left (2\right )}^2\,4{}\mathrm {i}-\Pi ^2\,{\ln \left (4\right )}^2\,5{}\mathrm {i}-{\ln \left (2\right )}^2\,{\ln \left (4\right )}^2\,4{}\mathrm {i}+4\,\Pi \,\ln \left (2\right )\,{\ln \left (4\right )}^2+8\,\Pi \,{\ln \left (2\right )}^2\,\ln \left (4\right )+\Pi ^2\,\ln \left (2\right )\,\ln \left (4\right )\,8{}\mathrm {i}\right )}{\left (\Pi ^2\,5{}\mathrm {i}+20\,\ln \left (2\right )\,\Pi -{\ln \left (2\right )}^2\,20{}\mathrm {i}-\mathrm {i}\right )\,\left (200\,\Pi ^3\,\ln \left (2\right )-800\,\Pi \,{\ln \left (2\right )}^3-40\,\Pi \,\ln \left (2\right )-\Pi ^2\,10{}\mathrm {i}+\Pi ^4\,25{}\mathrm {i}+{\ln \left (2\right )}^2\,40{}\mathrm {i}+{\ln \left (2\right )}^4\,400{}\mathrm {i}-\Pi ^2\,{\ln \left (2\right )}^2\,600{}\mathrm {i}+1{}\mathrm {i}\right )}}{x^4\,1{}\mathrm {i}+\left (-\Pi ^2\,10{}\mathrm {i}-40\,\ln \left (2\right )\,\Pi +{\ln \left (2\right )}^2\,40{}\mathrm {i}+2{}\mathrm {i}\right )\,x^3+\left (200\,\Pi ^3\,\ln \left (2\right )-800\,\Pi \,{\ln \left (2\right )}^3-40\,\Pi \,\ln \left (2\right )-\Pi ^2\,10{}\mathrm {i}+\Pi ^4\,25{}\mathrm {i}+{\ln \left (2\right )}^2\,40{}\mathrm {i}+{\ln \left (2\right )}^4\,400{}\mathrm {i}-\Pi ^2\,{\ln \left (2\right )}^2\,600{}\mathrm {i}+1{}\mathrm {i}\right )\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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