Optimal. Leaf size=64 \[ -\frac {5}{578} \log \left (2 x^2+2 x+1\right )-\frac {1}{34 (3 x+4)}+\frac {5}{289} \log (3 x+4)-\frac {\tan ^{-1}(2 x+1)}{6 (3 x+4)^2}+\frac {8}{867} \tan ^{-1}(2 x+1) \]
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Rubi [A] time = 0.07, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5045, 1982, 709, 800, 634, 617, 204, 628} \[ -\frac {5}{578} \log \left (2 x^2+2 x+1\right )-\frac {1}{34 (3 x+4)}+\frac {5}{289} \log (3 x+4)-\frac {\tan ^{-1}(2 x+1)}{6 (3 x+4)^2}+\frac {8}{867} \tan ^{-1}(2 x+1) \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 634
Rule 709
Rule 800
Rule 1982
Rule 5045
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(1+2 x)}{(4+3 x)^3} \, dx &=-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {1}{3} \int \frac {1}{(4+3 x)^2 \left (1+(1+2 x)^2\right )} \, dx\\ &=-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {1}{3} \int \frac {1}{(4+3 x)^2 \left (2+4 x+4 x^2\right )} \, dx\\ &=-\frac {1}{34 (4+3 x)}-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {1}{102} \int \frac {4-12 x}{(4+3 x) \left (2+4 x+4 x^2\right )} \, dx\\ &=-\frac {1}{34 (4+3 x)}-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {1}{102} \int \left (\frac {90}{17 (4+3 x)}-\frac {2 (7+30 x)}{17 \left (1+2 x+2 x^2\right )}\right ) \, dx\\ &=-\frac {1}{34 (4+3 x)}-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {5}{289} \log (4+3 x)-\frac {1}{867} \int \frac {7+30 x}{1+2 x+2 x^2} \, dx\\ &=-\frac {1}{34 (4+3 x)}-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {5}{289} \log (4+3 x)-\frac {5}{578} \int \frac {2+4 x}{1+2 x+2 x^2} \, dx+\frac {8}{867} \int \frac {1}{1+2 x+2 x^2} \, dx\\ &=-\frac {1}{34 (4+3 x)}-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {5}{289} \log (4+3 x)-\frac {5}{578} \log \left (1+2 x+2 x^2\right )-\frac {8}{867} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {1}{34 (4+3 x)}+\frac {8}{867} \tan ^{-1}(1+2 x)-\frac {\tan ^{-1}(1+2 x)}{6 (4+3 x)^2}+\frac {5}{289} \log (4+3 x)-\frac {5}{578} \log \left (1+2 x+2 x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 81, normalized size = 1.27 \[ \frac {-289 \tan ^{-1}(2 x+1)+(3 x+4) ((-15+8 i) (3 x+4) \log ((1+i) x+i)-(15+8 i) (3 x+4) \log (1+(1+i) x)+90 x \log (3 x+4)+120 \log (3 x+4)-51)}{1734 (3 x+4)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 77, normalized size = 1.20 \[ \frac {{\left (48 \, x^{2} + 128 \, x - 11\right )} \arctan \left (2 \, x + 1\right ) - 5 \, {\left (9 \, x^{2} + 24 \, x + 16\right )} \log \left (2 \, x^{2} + 2 \, x + 1\right ) + 10 \, {\left (9 \, x^{2} + 24 \, x + 16\right )} \log \left (3 \, x + 4\right ) - 51 \, x - 68}{578 \, {\left (9 \, x^{2} + 24 \, x + 16\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 0.84 \[ -\frac {2 \arctan \left (1+2 x \right )}{3 \left (8+6 x \right )^{2}}-\frac {1}{17 \left (8+6 x \right )}+\frac {5 \ln \left (8+6 x \right )}{289}-\frac {5 \ln \left (\left (1+2 x \right )^{2}+1\right )}{578}+\frac {8 \arctan \left (1+2 x \right )}{867} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 54, normalized size = 0.84 \[ -\frac {1}{34 \, {\left (3 \, x + 4\right )}} - \frac {\arctan \left (2 \, x + 1\right )}{6 \, {\left (3 \, x + 4\right )}^{2}} + \frac {8}{867} \, \arctan \left (2 \, x + 1\right ) - \frac {5}{578} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) + \frac {5}{289} \, \log \left (3 \, x + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 46, normalized size = 0.72 \[ \frac {5\,\ln \left (x+\frac {4}{3}\right )}{289}-\frac {5\,\ln \left (x^2+x+\frac {1}{2}\right )}{578}+\frac {8\,\mathrm {atan}\left (2\,x+1\right )}{867}-\frac {\frac {3\,x}{34}+\frac {\mathrm {atan}\left (2\,x+1\right )}{6}+\frac {2}{17}}{{\left (3\,x+4\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 223, normalized size = 3.48 \[ \frac {90 x^{2} \log {\left (x + \frac {4}{3} \right )}}{5202 x^{2} + 13872 x + 9248} - \frac {45 x^{2} \log {\left (2 x^{2} + 2 x + 1 \right )}}{5202 x^{2} + 13872 x + 9248} + \frac {48 x^{2} \operatorname {atan}{\left (2 x + 1 \right )}}{5202 x^{2} + 13872 x + 9248} + \frac {240 x \log {\left (x + \frac {4}{3} \right )}}{5202 x^{2} + 13872 x + 9248} - \frac {120 x \log {\left (2 x^{2} + 2 x + 1 \right )}}{5202 x^{2} + 13872 x + 9248} + \frac {128 x \operatorname {atan}{\left (2 x + 1 \right )}}{5202 x^{2} + 13872 x + 9248} - \frac {51 x}{5202 x^{2} + 13872 x + 9248} + \frac {160 \log {\left (x + \frac {4}{3} \right )}}{5202 x^{2} + 13872 x + 9248} - \frac {80 \log {\left (2 x^{2} + 2 x + 1 \right )}}{5202 x^{2} + 13872 x + 9248} - \frac {11 \operatorname {atan}{\left (2 x + 1 \right )}}{5202 x^{2} + 13872 x + 9248} - \frac {68}{5202 x^{2} + 13872 x + 9248} \]
Verification of antiderivative is not currently implemented for this CAS.
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