Optimal. Leaf size=35 \[ -\frac {15 x}{64 \left (16+x^2\right )^2}+\frac {19 x}{2048 \left (16+x^2\right )}+\frac {19 \tan ^{-1}\left (\frac {x}{4}\right )}{8192} \]
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Rubi [A]
time = 0.00, 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 = {393, 205, 209}
\begin {gather*} \frac {19 \text {ArcTan}\left (\frac {x}{4}\right )}{8192}+\frac {19 x}{2048 \left (x^2+16\right )}-\frac {15 x}{64 \left (x^2+16\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 209
Rule 393
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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 25, normalized size = 0.71
method | result | size |
default | \(\frac {\frac {19}{2048} x^{3}-\frac {11}{128} x}{\left (x^{2}+16\right )^{2}}+\frac {19 \arctan \left (\frac {x}{4}\right )}{8192}\) | \(25\) |
risch | \(\frac {\frac {19}{2048} x^{3}-\frac {11}{128} x}{\left (x^{2}+16\right )^{2}}+\frac {19 \arctan \left (\frac {x}{4}\right )}{8192}\) | \(25\) |
meijerg | \(\frac {x \left (\frac {3 x^{2}}{16}+5\right )}{32768 \left (\frac {x^{2}}{16}+1\right )^{2}}+\frac {19 \arctan \left (\frac {x}{4}\right )}{8192}-\frac {x \left (-\frac {3 x^{2}}{16}+3\right )}{6144 \left (\frac {x^{2}}{16}+1\right )^{2}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 30, normalized size = 0.86 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.70, size = 39, normalized size = 1.11 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 27, normalized size = 0.77 \begin {gather*} \frac {19 x^{3} - 176 x}{2048 x^{4} + 65536 x^{2} + 524288} + \frac {19 \operatorname {atan}{\left (\frac {x}{4} \right )}}{8192} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 25, normalized size = 0.71 \begin {gather*} \frac {19 \, x^{3} - 176 \, x}{2048 \, {\left (x^{2} + 16\right )}^{2}} + \frac {19}{8192} \, \arctan \left (\frac {1}{4} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 30, normalized size = 0.86 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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