Optimal. Leaf size=84 \[ \frac {x}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \text {ArcTan}(a x)}{32 a^2 c^3}-\frac {\text {ArcTan}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5050, 205, 211}
\begin {gather*} -\frac {\text {ArcTan}(a x)}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \text {ArcTan}(a x)}{32 a^2 c^3}+\frac {3 x}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {x}{16 a c^3 \left (a^2 x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 5050
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac {\tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx}{4 a}\\ &=\frac {x}{16 a c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=\frac {x}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x}{32 a c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{32 a c^2}\\ &=\frac {x}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3}-\frac {\tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.65 \begin {gather*} \frac {a x \left (5+3 a^2 x^2\right )+\left (-5+6 a^2 x^2+3 a^4 x^4\right ) \text {ArcTan}(a x)}{32 c^3 \left (a+a^3 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 68, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {-\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {a x}{4 \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 a x}{8 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )}{8}}{4 c^{3}}}{a^{2}}\) | \(68\) |
default | \(\frac {-\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {a x}{4 \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 a x}{8 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )}{8}}{4 c^{3}}}{a^{2}}\) | \(68\) |
risch | \(\frac {i \ln \left (i a x +1\right )}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {i \left (8 \ln \left (-i a x +1\right )+3 \ln \left (-a x +i\right ) a^{4} x^{4}+6 \ln \left (-a x +i\right ) a^{2} x^{2}+3 \ln \left (-a x +i\right )-3 \ln \left (a x +i\right ) a^{4} x^{4}-6 \ln \left (a x +i\right ) a^{2} x^{2}-3 \ln \left (a x +i\right )+6 i a^{3} x^{3}+10 i a x \right )}{64 a^{2} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}\) | \(161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 86, normalized size = 1.02 \begin {gather*} \frac {\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}} + \frac {3 \, \arctan \left (a x\right )}{a c^{2}}}{32 \, a c} - \frac {\arctan \left (a x\right )}{4 \, {\left (a^{2} c x^{2} + c\right )}^{2} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 69, normalized size = 0.82 \begin {gather*} \frac {3 \, a^{3} x^{3} + 5 \, a x + {\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 5\right )} \arctan \left (a x\right )}{32 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (75) = 150\).
time = 0.72, size = 209, normalized size = 2.49 \begin {gather*} \begin {cases} \frac {3 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{32 a^{6} c^{3} x^{4} + 64 a^{4} c^{3} x^{2} + 32 a^{2} c^{3}} + \frac {3 a^{3} x^{3}}{32 a^{6} c^{3} x^{4} + 64 a^{4} c^{3} x^{2} + 32 a^{2} c^{3}} + \frac {6 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{32 a^{6} c^{3} x^{4} + 64 a^{4} c^{3} x^{2} + 32 a^{2} c^{3}} + \frac {5 a x}{32 a^{6} c^{3} x^{4} + 64 a^{4} c^{3} x^{2} + 32 a^{2} c^{3}} - \frac {5 \operatorname {atan}{\left (a x \right )}}{32 a^{6} c^{3} x^{4} + 64 a^{4} c^{3} x^{2} + 32 a^{2} c^{3}} & \text {for}\: a \neq 0 \\0 & \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 = 0.48, size = 103, normalized size = 1.23 \begin {gather*} \frac {\frac {5\,x}{32\,a}+\frac {a\,x^3}{4}-\frac {\mathrm {atan}\left (a\,x\right )}{4\,a^2}-\frac {x^2\,\mathrm {atan}\left (a\,x\right )}{4}+\frac {3\,a^3\,x^5}{32}}{a^6\,c^3\,x^6+3\,a^4\,c^3\,x^4+3\,a^2\,c^3\,x^2+c^3}+\frac {3\,\mathrm {atan}\left (\frac {a^2\,x}{\sqrt {a^2}}\right )}{32\,a\,c^3\,\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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