Optimal. Leaf size=35 \[ \frac {19 x}{2048 \left (x^2+16\right )}-\frac {15 x}{64 \left (x^2+16\right )^2}+\frac {19 \tan ^{-1}\left (\frac {x}{4}\right )}{8192} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {385, 199, 203} \[ \frac {19 x}{2048 \left (x^2+16\right )}-\frac {15 x}{64 \left (x^2+16\right )^2}+\frac {19 \tan ^{-1}\left (\frac {x}{4}\right )}{8192} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 385
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (16+x^2\right )^3} \, dx &=-\frac {15 x}{64 \left (16+x^2\right )^2}+\frac {19}{64} \int \frac {1}{\left (16+x^2\right )^2} \, dx\\ &=-\frac {15 x}{64 \left (16+x^2\right )^2}+\frac {19 x}{2048 \left (16+x^2\right )}+\frac {19 \int \frac {1}{16+x^2} \, dx}{2048}\\ &=-\frac {15 x}{64 \left (16+x^2\right )^2}+\frac {19 x}{2048 \left (16+x^2\right )}+\frac {19 \tan ^{-1}\left (\frac {x}{4}\right )}{8192}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ \frac {19 x}{2048 \left (x^2+16\right )}-\frac {15 x}{64 \left (x^2+16\right )^2}+\frac {19 \tan ^{-1}\left (\frac {x}{4}\right )}{8192} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 39, normalized size = 1.11 \[ \frac {76 \, x^{3} + 19 \, {\left (x^{4} + 32 \, x^{2} + 256\right )} \arctan \left (\frac {1}{4} \, x\right ) - 704 \, x}{8192 \, {\left (x^{4} + 32 \, x^{2} + 256\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 25, normalized size = 0.71 \[ \frac {19 \, x^{3} - 176 \, x}{2048 \, {\left (x^{2} + 16\right )}^{2}} + \frac {19}{8192} \, \arctan \left (\frac {1}{4} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.71 \[ \frac {19 \arctan \left (\frac {x}{4}\right )}{8192}+\frac {\frac {19}{2048} x^{3}-\frac {11}{128} x}{\left (x^{2}+16\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 30, normalized size = 0.86 \[ \frac {19 \, x^{3} - 176 \, x}{2048 \, {\left (x^{4} + 32 \, x^{2} + 256\right )}} + \frac {19}{8192} \, \arctan \left (\frac {1}{4} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 30, normalized size = 0.86 \[ \frac {19\,\mathrm {atan}\left (\frac {x}{4}\right )}{8192}-\frac {\frac {11\,x}{128}-\frac {19\,x^3}{2048}}{x^4+32\,x^2+256} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 27, normalized size = 0.77 \[ \frac {19 x^{3} - 176 x}{2048 x^{4} + 65536 x^{2} + 524288} + \frac {19 \operatorname {atan}{\left (\frac {x}{4} \right )}}{8192} \]
Verification of antiderivative is not currently implemented for this CAS.
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