Optimal. Leaf size=221 \[ \frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}-\frac {b (2 d-e) x^3}{24 c}+\frac {b e x^3}{18 c}-\frac {b (2 d-3 e) \text {ArcTan}(c x)}{8 c^4}+\frac {2 b e \text {ArcTan}(c x)}{3 c^4}+\frac {e x^2 (a+b \text {ArcTan}(c x))}{4 c^2}-\frac {1}{8} e x^4 (a+b \text {ArcTan}(c x))+\frac {b e x \log \left (1+c^2 x^2\right )}{4 c^3}-\frac {b e x^3 \log \left (1+c^2 x^2\right )}{12 c}-\frac {e (a+b \text {ArcTan}(c x)) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 (a+b \text {ArcTan}(c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \]
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Rubi [A]
time = 0.18, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2504, 2442,
45, 5139, 470, 327, 209, 2521, 2498, 2505, 308} \begin {gather*} \frac {1}{4} x^4 (a+b \text {ArcTan}(c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac {e x^2 (a+b \text {ArcTan}(c x))}{4 c^2}-\frac {e \log \left (c^2 x^2+1\right ) (a+b \text {ArcTan}(c x))}{4 c^4}-\frac {1}{8} e x^4 (a+b \text {ArcTan}(c x))-\frac {b (2 d-3 e) \text {ArcTan}(c x)}{8 c^4}+\frac {2 b e \text {ArcTan}(c x)}{3 c^4}+\frac {b x (2 d-3 e)}{8 c^3}-\frac {2 b e x}{3 c^3}-\frac {b e x^3 \log \left (c^2 x^2+1\right )}{12 c}+\frac {b e x \log \left (c^2 x^2+1\right )}{4 c^3}-\frac {b x^3 (2 d-e)}{24 c}+\frac {b e x^3}{18 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 209
Rule 308
Rule 327
Rule 470
Rule 2442
Rule 2498
Rule 2504
Rule 2505
Rule 2521
Rule 5139
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx &=\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-(b c) \int \left (\frac {x^2 \left (2 e+c^2 (2 d-e) x^2\right )}{8 c^2 \left (1+c^2 x^2\right )}+\frac {e \left (-1+c^2 x^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}\right ) \, dx\\ &=\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \int \frac {x^2 \left (2 e+c^2 (2 d-e) x^2\right )}{1+c^2 x^2} \, dx}{8 c}-\frac {(b e) \int \left (-1+c^2 x^2\right ) \log \left (1+c^2 x^2\right ) \, dx}{4 c^3}\\ &=-\frac {b (2 d-e) x^3}{24 c}+\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+\frac {(b (2 d-3 e)) \int \frac {x^2}{1+c^2 x^2} \, dx}{8 c}-\frac {(b e) \int \left (-\log \left (1+c^2 x^2\right )+c^2 x^2 \log \left (1+c^2 x^2\right )\right ) \, dx}{4 c^3}\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {b (2 d-e) x^3}{24 c}+\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {1}{1+c^2 x^2} \, dx}{8 c^3}+\frac {(b e) \int \log \left (1+c^2 x^2\right ) \, dx}{4 c^3}-\frac {(b e) \int x^2 \log \left (1+c^2 x^2\right ) \, dx}{4 c}\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {b (2 d-e) x^3}{24 c}-\frac {b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {b e x \log \left (1+c^2 x^2\right )}{4 c^3}-\frac {b e x^3 \log \left (1+c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {(b e) \int \frac {x^2}{1+c^2 x^2} \, dx}{2 c}+\frac {1}{6} (b c e) \int \frac {x^4}{1+c^2 x^2} \, dx\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {b e x}{2 c^3}-\frac {b (2 d-e) x^3}{24 c}-\frac {b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {b e x \log \left (1+c^2 x^2\right )}{4 c^3}-\frac {b e x^3 \log \left (1+c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+\frac {(b e) \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3}+\frac {1}{6} (b c e) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}-\frac {b (2 d-e) x^3}{24 c}+\frac {b e x^3}{18 c}-\frac {b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac {b e \tan ^{-1}(c x)}{2 c^4}+\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {b e x \log \left (1+c^2 x^2\right )}{4 c^3}-\frac {b e x^3 \log \left (1+c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+\frac {(b e) \int \frac {1}{1+c^2 x^2} \, dx}{6 c^3}\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}-\frac {b (2 d-e) x^3}{24 c}+\frac {b e x^3}{18 c}-\frac {b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac {2 b e \tan ^{-1}(c x)}{3 c^4}+\frac {e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {b e x \log \left (1+c^2 x^2\right )}{4 c^3}-\frac {b e x^3 \log \left (1+c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 164, normalized size = 0.74 \begin {gather*} \frac {c x \left (18 a c^3 d x^3-6 b d \left (-3+c^2 x^2\right )-9 a c e x \left (-2+c^2 x^2\right )+b e \left (-75+7 c^2 x^2\right )\right )-6 e \left (b c x \left (-3+c^2 x^2\right )+a \left (3-3 c^4 x^4\right )\right ) \log \left (1+c^2 x^2\right )+3 b \text {ArcTan}(c x) \left (e \left (25+6 c^2 x^2-3 c^4 x^4\right )+6 d \left (-1+c^4 x^4\right )+6 e \left (-1+c^4 x^4\right ) \log \left (1+c^2 x^2\right )\right )}{72 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.30, size = 3897, normalized size = 17.63
method | result | size |
default | \(\text {Expression too large to display}\) | \(3897\) |
risch | \(\text {Expression too large to display}\) | \(22188\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 227, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, a d x^{4} + \frac {1}{72} \, b c {\left (\frac {7 \, c^{2} x^{3} - 6 \, {\left (c^{2} x^{3} - 3 \, x\right )} \log \left (c^{2} x^{2} + 1\right ) - 75 \, x}{c^{4}} + \frac {75 \, \arctan \left (c x\right )}{c^{5}}\right )} e + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b \arctan \left (c x\right ) e + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} a e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.37, size = 179, normalized size = 0.81 \begin {gather*} \frac {18 \, a c^{4} d x^{4} - 6 \, b c^{3} d x^{3} + 18 \, b c d x + 3 \, {\left (6 \, b c^{4} d x^{4} - 6 \, b d - {\left (3 \, b c^{4} x^{4} - 6 \, b c^{2} x^{2} - 25 \, b\right )} e\right )} \arctan \left (c x\right ) - {\left (9 \, a c^{4} x^{4} - 7 \, b c^{3} x^{3} - 18 \, a c^{2} x^{2} + 75 \, b c x\right )} e + 6 \, {\left (3 \, {\left (b c^{4} x^{4} - b\right )} \arctan \left (c x\right ) e + {\left (3 \, a c^{4} x^{4} - b c^{3} x^{3} + 3 \, b c x - 3 \, a\right )} e\right )} \log \left (c^{2} x^{2} + 1\right )}{72 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.32, size = 279, normalized size = 1.26 \begin {gather*} \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{4} \log {\left (c^{2} x^{2} + 1 \right )}}{4} - \frac {a e x^{4}}{8} + \frac {a e x^{2}}{4 c^{2}} - \frac {a e \log {\left (c^{2} x^{2} + 1 \right )}}{4 c^{4}} + \frac {b d x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b e x^{4} \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{4} - \frac {b e x^{4} \operatorname {atan}{\left (c x \right )}}{8} - \frac {b d x^{3}}{12 c} - \frac {b e x^{3} \log {\left (c^{2} x^{2} + 1 \right )}}{12 c} + \frac {7 b e x^{3}}{72 c} + \frac {b e x^{2} \operatorname {atan}{\left (c x \right )}}{4 c^{2}} + \frac {b d x}{4 c^{3}} + \frac {b e x \log {\left (c^{2} x^{2} + 1 \right )}}{4 c^{3}} - \frac {25 b e x}{24 c^{3}} - \frac {b d \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b e \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} + \frac {25 b e \operatorname {atan}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.78, size = 297, normalized size = 1.34 \begin {gather*} \frac {a\,d\,x^4}{4}-\frac {a\,e\,x^4}{8}+\frac {b\,d\,x}{4\,c^3}-\frac {25\,b\,e\,x}{24\,c^3}+\frac {b\,d\,x^4\,\mathrm {atan}\left (c\,x\right )}{4}-\frac {b\,e\,x^4\,\mathrm {atan}\left (c\,x\right )}{8}-\frac {a\,e\,\ln \left (c^2\,x^2+1\right )}{4\,c^4}+\frac {a\,e\,x^2}{4\,c^2}-\frac {b\,d\,x^3}{12\,c}-\frac {b\,d\,\mathrm {atan}\left (\frac {6\,b\,c\,d\,x}{6\,b\,d-25\,b\,e}-\frac {25\,b\,c\,e\,x}{6\,b\,d-25\,b\,e}\right )}{4\,c^4}+\frac {7\,b\,e\,x^3}{72\,c}+\frac {25\,b\,e\,\mathrm {atan}\left (\frac {6\,b\,c\,d\,x}{6\,b\,d-25\,b\,e}-\frac {25\,b\,c\,e\,x}{6\,b\,d-25\,b\,e}\right )}{24\,c^4}+\frac {a\,e\,x^4\,\ln \left (c^2\,x^2+1\right )}{4}+\frac {b\,e\,x\,\ln \left (c^2\,x^2+1\right )}{4\,c^3}-\frac {b\,e\,\mathrm {atan}\left (c\,x\right )\,\ln \left (c^2\,x^2+1\right )}{4\,c^4}+\frac {b\,e\,x^2\,\mathrm {atan}\left (c\,x\right )}{4\,c^2}+\frac {b\,e\,x^4\,\mathrm {atan}\left (c\,x\right )\,\ln \left (c^2\,x^2+1\right )}{4}-\frac {b\,e\,x^3\,\ln \left (c^2\,x^2+1\right )}{12\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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