Optimal. Leaf size=31 \[ -x+\frac {3}{5} x^2 (4+x) \left (-e^x+x\right ) \log \left (\left (\frac {4}{x}+x\right )^2\right ) \]
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Rubi [B] time = 4.49, antiderivative size = 86, normalized size of antiderivative = 2.77, number of steps used = 152, number of rules used = 38, integrand size = 120, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {6725, 203, 321, 266, 43, 302, 2528, 2523, 12, 388, 2526, 446, 72, 4848, 2391, 4920, 4854, 2402, 2315, 2525, 444, 2524, 2418, 2394, 260, 2416, 2390, 2301, 2393, 459, 77, 6688, 517, 2194, 2176, 2178, 2196, 2554} \begin {gather*} -\frac {12}{5} e^x x^2 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 72
Rule 77
Rule 203
Rule 260
Rule 266
Rule 302
Rule 321
Rule 388
Rule 444
Rule 446
Rule 459
Rule 517
Rule 2176
Rule 2178
Rule 2194
Rule 2196
Rule 2301
Rule 2315
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2402
Rule 2416
Rule 2418
Rule 2523
Rule 2524
Rule 2525
Rule 2526
Rule 2528
Rule 2554
Rule 4848
Rule 4854
Rule 4920
Rule 6688
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{4+x^2}-\frac {101 x^2}{5 \left (4+x^2\right )}-\frac {24 x^3}{5 \left (4+x^2\right )}+\frac {24 x^4}{5 \left (4+x^2\right )}+\frac {6 x^5}{5 \left (4+x^2\right )}+\frac {144 x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}+\frac {48 x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}+\frac {36 x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}+\frac {12 x^5 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}-\frac {3 e^x x \left (-32-8 x+8 x^2+2 x^3+32 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+28 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+12 x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+7 x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right )}{5 \left (4+x^2\right )}\right ) \, dx\\ &=-\left (\frac {3}{5} \int \frac {e^x x \left (-32-8 x+8 x^2+2 x^3+32 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+28 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+12 x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+7 x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right )}{4+x^2} \, dx\right )+\frac {6}{5} \int \frac {x^5}{4+x^2} \, dx+\frac {12}{5} \int \frac {x^5 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx-4 \int \frac {1}{4+x^2} \, dx-\frac {24}{5} \int \frac {x^3}{4+x^2} \, dx+\frac {24}{5} \int \frac {x^4}{4+x^2} \, dx+\frac {36}{5} \int \frac {x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx+\frac {48}{5} \int \frac {x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx-\frac {101}{5} \int \frac {x^2}{4+x^2} \, dx+\frac {144}{5} \int \frac {x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx\\ &=-\frac {101 x}{5}-2 \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3}{5} \int \frac {e^x x \left (2 \left (-16-4 x+4 x^2+x^3\right )+\left (32+28 x+12 x^2+7 x^3+x^4\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right )}{4+x^2} \, dx+\frac {3}{5} \operatorname {Subst}\left (\int \frac {x^2}{4+x} \, dx,x,x^2\right )+\frac {12}{5} \int \left (-4 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {16 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx-\frac {12}{5} \operatorname {Subst}\left (\int \frac {x}{4+x} \, dx,x,x^2\right )+\frac {24}{5} \int \left (-4+x^2+\frac {16}{4+x^2}\right ) \, dx+\frac {36}{5} \int \left (-4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {16 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx+\frac {48}{5} \int \left (x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {4 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx+\frac {144}{5} \int \left (\log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx+\frac {404}{5} \int \frac {1}{4+x^2} \, dx\\ &=-\frac {197 x}{5}+\frac {8 x^3}{5}+\frac {192}{5} \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3}{5} \int \left (\frac {2 e^x (-2+x) x (2+x) (4+x)}{4+x^2}+e^x x \left (8+7 x+x^2\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right ) \, dx+\frac {3}{5} \operatorname {Subst}\left (\int \left (-4+x+\frac {16}{4+x}\right ) \, dx,x,x^2\right )+\frac {12}{5} \int x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \, dx-\frac {12}{5} \operatorname {Subst}\left (\int \left (1-\frac {4}{4+x}\right ) \, dx,x,x^2\right )+\frac {36}{5} \int x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \, dx+\frac {384}{5} \int \frac {1}{4+x^2} \, dx\\ &=-\frac {197 x}{5}-\frac {24 x^2}{5}+\frac {8 x^3}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} \int \frac {2 x^3 \left (-4+x^2\right )}{4+x^2} \, dx-\frac {3}{5} \int e^x x \left (8+7 x+x^2\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \, dx-\frac {6}{5} \int \frac {e^x (-2+x) x (2+x) (4+x)}{4+x^2} \, dx-\frac {12}{5} \int \frac {2 x^2 \left (-4+x^2\right )}{4+x^2} \, dx\\ &=-\frac {197 x}{5}-\frac {24 x^2}{5}+\frac {8 x^3}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} \int \frac {2 e^x x (4+x) \left (-4+x^2\right )}{4+x^2} \, dx-\frac {6}{5} \int \frac {x^3 \left (-4+x^2\right )}{4+x^2} \, dx-\frac {6}{5} \int \frac {e^x x (4+x) \left (-4+x^2\right )}{4+x^2} \, dx-\frac {24}{5} \int \frac {x^2 \left (-4+x^2\right )}{4+x^2} \, dx\\ &=-\frac {197 x}{5}-\frac {24 x^2}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} \operatorname {Subst}\left (\int \frac {(-4+x) x}{4+x} \, dx,x,x^2\right )+\frac {6}{5} \int \frac {e^x x (4+x) \left (-4+x^2\right )}{4+x^2} \, dx-\frac {6}{5} \int \left (-8 e^x+4 e^x x+e^x x^2+\frac {32 e^x (1-x)}{4+x^2}\right ) \, dx+\frac {192}{5} \int \frac {x^2}{4+x^2} \, dx\\ &=-x-\frac {24 x^2}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} \operatorname {Subst}\left (\int \left (-8+x+\frac {32}{4+x}\right ) \, dx,x,x^2\right )-\frac {6}{5} \int e^x x^2 \, dx+\frac {6}{5} \int \left (-8 e^x+4 e^x x+e^x x^2+\frac {32 e^x (1-x)}{4+x^2}\right ) \, dx-\frac {24}{5} \int e^x x \, dx+\frac {48 \int e^x \, dx}{5}-\frac {192}{5} \int \frac {e^x (1-x)}{4+x^2} \, dx-\frac {768}{5} \int \frac {1}{4+x^2} \, dx\\ &=\frac {48 e^x}{5}-x-\frac {24 e^x x}{5}-\frac {6 e^x x^2}{5}-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {6}{5} \int e^x x^2 \, dx+\frac {12}{5} \int e^x x \, dx+\frac {24 \int e^x \, dx}{5}+\frac {24}{5} \int e^x x \, dx-\frac {48 \int e^x \, dx}{5}+\frac {192}{5} \int \frac {e^x (1-x)}{4+x^2} \, dx-\frac {192}{5} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{4}\right ) e^x}{2 i-x}-\frac {\left (\frac {1}{2}-\frac {i}{4}\right ) e^x}{2 i+x}\right ) \, dx\\ &=\frac {24 e^x}{5}-x+\frac {12 e^x x}{5}-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\left (-\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i+x} \, dx-\frac {12 \int e^x \, dx}{5}-\frac {12}{5} \int e^x x \, dx-\frac {24 \int e^x \, dx}{5}-\left (\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i-x} \, dx+\frac {192}{5} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{4}\right ) e^x}{2 i-x}-\frac {\left (\frac {1}{2}-\frac {i}{4}\right ) e^x}{2 i+x}\right ) \, dx\\ &=-\frac {12 e^x}{5}-x+\left (\frac {96}{5}+\frac {48 i}{5}\right ) e^{2 i} \text {Ei}(-2 i+x)+\left (\frac {96}{5}-\frac {48 i}{5}\right ) e^{-2 i} \text {Ei}(2 i+x)-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\left (-\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i+x} \, dx+\frac {12 \int e^x \, dx}{5}+\left (\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i-x} \, dx\\ &=-x-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 35, normalized size = 1.13 \begin {gather*} \frac {1}{5} \left (-5 x+3 x^2 (4+x) \left (-e^x+x\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 43, normalized size = 1.39 \begin {gather*} \frac {3}{5} \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{3} + 4 \, x^{2}\right )} e^{x}\right )} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 88, normalized size = 2.84 \begin {gather*} \frac {3}{5} \, x^{4} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - \frac {3}{5} \, x^{3} e^{x} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) + \frac {12}{5} \, x^{3} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - \frac {12}{5} \, x^{2} e^{x} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.55, size = 1157, normalized size = 37.32
method | result | size |
risch | \(\frac {6 x^{3} {\mathrm e}^{x} \ln \relax (x )}{5}-x +\frac {24 x^{2} {\mathrm e}^{x} \ln \relax (x )}{5}-\frac {24 x^{3} \ln \relax (x )}{5}+\frac {12 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}}{5}-\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) {\mathrm e}^{x}}{10}-\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) {\mathrm e}^{x}}{5}+\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) {\mathrm e}^{x}}{10}+\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) {\mathrm e}^{x}}{5}-\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right ) \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{2} {\mathrm e}^{x}}{5}-\frac {6 x^{4} \ln \relax (x )}{5}-\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{3}}{5}+\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i x^{2}\right )^{3}}{10}+\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}}{5}-\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{3}}{10}-\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{3}}{5}-\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{3}}{10}+\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{10}+\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{3} {\mathrm e}^{x}}{10}+\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{3} {\mathrm e}^{x}}{5}+\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{3} {\mathrm e}^{x}}{10}+\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{3} {\mathrm e}^{x}}{5}-\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )}{10}-\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}}{5}-\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )}{5}+\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right ) \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{2}}{5}+\frac {12 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right ) \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{2}}{5}-\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}}{10}-\frac {12 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )\right ) \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right )^{2} {\mathrm e}^{x}}{5}-\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}}{10}-\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2} {\mathrm e}^{x}}{10}-\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}{10}-\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}{5}-\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2} {\mathrm e}^{x}}{5}-\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}}{5}+\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}}{5}+\left (\frac {6 x^{4}}{5}+\frac {24 x^{3}}{5}-\frac {24 \,{\mathrm e}^{x} x^{2}}{5}-\frac {6 \,{\mathrm e}^{x} x^{3}}{5}\right ) \ln \left (x^{2}+4\right )+\frac {3 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) {\mathrm e}^{x}}{10}+\frac {6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) {\mathrm e}^{x}}{5}+\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{5}-\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{5}-\frac {12 i \pi \,x^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{5}+\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2}}{10}+\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (i \left (x^{2}+4\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2}}{5}+\frac {3 i \pi \,x^{4} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}{10}+\frac {6 i \pi \,x^{3} \mathrm {csgn}\left (\frac {i \left (x^{2}+4\right )^{2}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}{5}\) | \(1157\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 71, normalized size = 2.29 \begin {gather*} \frac {6}{5} \, {\left (x^{3} + 4 \, x^{2}\right )} e^{x} \log \relax (x) + \frac {6}{5} \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{3} + 4 \, x^{2}\right )} e^{x} - 16\right )} \log \left (x^{2} + 4\right ) - \frac {6}{5} \, {\left (x^{4} + 4 \, x^{3}\right )} \log \relax (x) - x + \frac {96}{5} \, \log \left (x^{2} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 46, normalized size = 1.48 \begin {gather*} \ln \left (\frac {x^4+8\,x^2+16}{x^2}\right )\,\left (\frac {12\,x^3}{5}-{\mathrm {e}}^x\,\left (\frac {3\,x^3}{5}+\frac {12\,x^2}{5}\right )+\frac {3\,x^4}{5}\right )-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.62, size = 76, normalized size = 2.45 \begin {gather*} - x + \left (\frac {3 x^{4}}{5} + \frac {12 x^{3}}{5}\right ) \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )} + \frac {\left (- 3 x^{3} \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )} - 12 x^{2} \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )}\right ) e^{x}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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