Optimal. Leaf size=183 \[ -\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\text {ArcTan}(a x)}{3 c^3 x^3}+\frac {3 a^2 \text {ArcTan}(a x)}{c^3 x}+\frac {a^4 x \text {ArcTan}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^4 x \text {ArcTan}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {35 a^3 \text {ArcTan}(a x)^2}{16 c^3}-\frac {10 a^3 \log (x)}{3 c^3}+\frac {5 a^3 \log \left (1+a^2 x^2\right )}{3 c^3} \]
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Rubi [A]
time = 0.51, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 38, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5086, 5038,
4946, 272, 46, 36, 29, 31, 5004, 5012, 267, 5016} \begin {gather*} \frac {35 a^3 \text {ArcTan}(a x)^2}{16 c^3}-\frac {10 a^3 \log (x)}{3 c^3}+\frac {3 a^2 \text {ArcTan}(a x)}{c^3 x}+\frac {11 a^4 x \text {ArcTan}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \text {ArcTan}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^3}{16 c^3 \left (a^2 x^2+1\right )}+\frac {a^3}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 a^3 \log \left (a^2 x^2+1\right )}{3 c^3}-\frac {\text {ArcTan}(a x)}{3 c^3 x^3}-\frac {a}{6 c^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 267
Rule 272
Rule 4946
Rule 5004
Rule 5012
Rule 5016
Rule 5038
Rule 5086
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\int \frac {\tan ^{-1}(a x)}{x^4} \, dx}{c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac {\left (3 a^4\right ) \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c^3}+\frac {a^4 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{c^2}-\frac {\left (3 a^5\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c^3}-\frac {a^4 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{c^2}+\frac {a^5 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {a \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )}{6 c^3}-\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}\right )\\ &=\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {a \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )}{6 c^3}-\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}\right )\\ &=-\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^2}{16 c^3}-\frac {a^3 \log (x)}{3 c^3}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c^3}-\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\right )+\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^2}{16 c^3}-\frac {4 a^3 \log (x)}{3 c^3}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^3 \log (x)}{c^3}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 142, normalized size = 0.78 \begin {gather*} \frac {2 \left (-8+56 a^2 x^2+175 a^4 x^4+105 a^6 x^6\right ) \text {ArcTan}(a x)+105 a^3 x^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2+a x \left (-8+20 a^2 x^2+25 a^4 x^4-160 \left (a x+a^3 x^3\right )^2 \log (x)+80 \left (a x+a^3 x^3\right )^2 \log \left (1+a^2 x^2\right )\right )}{48 c^3 x^3 \left (1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 161, normalized size = 0.88
method | result | size |
derivativedivides | \(a^{3} \left (\frac {11 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )}{c^{3} a x}-\frac {-\frac {33}{2 \left (a^{2} x^{2}+1\right )}-40 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )+\frac {105 \arctan \left (a x \right )^{2}}{2}}{24 c^{3}}\right )\) | \(161\) |
default | \(a^{3} \left (\frac {11 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )}{c^{3} a x}-\frac {-\frac {33}{2 \left (a^{2} x^{2}+1\right )}-40 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )+\frac {105 \arctan \left (a x \right )^{2}}{2}}{24 c^{3}}\right )\) | \(161\) |
risch | \(-\frac {35 a^{3} \ln \left (i a x +1\right )^{2}}{64 c^{3}}+\frac {\left (105 a^{7} x^{7} \ln \left (-i a x +1\right )-210 i x^{6} a^{6}+210 a^{5} x^{5} \ln \left (-i a x +1\right )-350 i x^{4} a^{4}+105 a^{3} x^{3} \ln \left (-i a x +1\right )-112 i a^{2} x^{2}+16 i\right ) \ln \left (i a x +1\right )}{96 x^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {-105 a^{7} x^{7} \ln \left (-i a x +1\right )^{2}+320 \ln \left (3 a^{2} x^{2}+3\right ) a^{7} x^{7}-640 \ln \left (x \right ) a^{7} x^{7}+224 i a^{2} x^{2} \ln \left (-i a x +1\right )-210 a^{5} x^{5} \ln \left (-i a x +1\right )^{2}+640 \ln \left (3 a^{2} x^{2}+3\right ) a^{5} x^{5}-1280 \ln \left (x \right ) a^{5} x^{5}+420 i a^{6} x^{6} \ln \left (-i a x +1\right )+100 a^{5} x^{5}-105 a^{3} x^{3} \ln \left (-i a x +1\right )^{2}+320 \ln \left (3 a^{2} x^{2}+3\right ) a^{3} x^{3}-640 \ln \left (x \right ) a^{3} x^{3}+700 i a^{4} \ln \left (-i a x +1\right ) x^{4}+80 a^{3} x^{3}-32 i \ln \left (-i a x +1\right )-32 a x}{192 x^{3} \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2}}\) | \(373\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 223, normalized size = 1.22 \begin {gather*} \frac {1}{24} \, {\left (\frac {105 \, a^{3} \arctan \left (a x\right )}{c^{3}} + \frac {105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8}{a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (25 \, a^{4} x^{4} + 20 \, a^{2} x^{2} - 105 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 80 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (x\right ) - 8\right )} a}{48 \, {\left (a^{4} c^{3} x^{6} + 2 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 179, normalized size = 0.98 \begin {gather*} \frac {25 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 105 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 8 \, a x + 2 \, {\left (105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8\right )} \arctan \left (a x\right ) + 80 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (x\right )}{48 \, {\left (a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{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. 724 vs.
\(2 (177) = 354\).
time = 2.14, size = 724, normalized size = 3.96 \begin {gather*} \begin {cases} - \frac {160 a^{7} x^{7} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{7} x^{7} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{7} x^{7} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{6} x^{6} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {320 a^{5} x^{5} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {160 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {25 a^{5} x^{5}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {350 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {160 a^{3} x^{3} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {20 a^{3} x^{3}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {112 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {8 a x}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {16 \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{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 = 0.59, size = 163, normalized size = 0.89 \begin {gather*} \frac {\frac {25\,a^5\,x^4}{2}+10\,a^3\,x^2-4\,a}{24\,a^4\,c^3\,x^6+48\,a^2\,c^3\,x^4+24\,c^3\,x^2}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {7\,x^2}{3\,c^3}-\frac {1}{3\,a^2\,c^3}+\frac {175\,a^2\,x^4}{24\,c^3}+\frac {35\,a^4\,x^6}{8\,c^3}\right )}{2\,x^5+\frac {x^3}{a^2}+a^2\,x^7}+\frac {5\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c^3}-\frac {10\,a^3\,\ln \left (x\right )}{3\,c^3}+\frac {35\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{16\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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