Integrand size = 14, antiderivative size = 115 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=-\frac {3 b c}{10 x^2}-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} \sqrt {3} b c^{5/3} \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )+\frac {1}{10} b c^{5/3} \log \left (1+c^{2/3} x^2\right )-\frac {1}{20} b c^{5/3} \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right ) \]
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Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4946, 281, 331, 298, 31, 648, 631, 210, 642} \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} \sqrt {3} b c^{5/3} \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )+\frac {1}{10} b c^{5/3} \log \left (c^{2/3} x^2+1\right )-\frac {1}{20} b c^{5/3} \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )-\frac {3 b c}{10 x^2} \]
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Rule 31
Rule 210
Rule 281
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rule 4946
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{5} (3 b c) \int \frac {1}{x^3 \left (1+c^2 x^6\right )} \, dx \\ & = -\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} (3 b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x^3\right )} \, dx,x,x^2\right ) \\ & = -\frac {3 b c}{10 x^2}-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}-\frac {1}{10} \left (3 b c^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^3} \, dx,x,x^2\right ) \\ & = -\frac {3 b c}{10 x^2}-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} \left (b c^{7/3}\right ) \text {Subst}\left (\int \frac {1}{1+c^{2/3} x} \, dx,x,x^2\right )-\frac {1}{10} \left (b c^{7/3}\right ) \text {Subst}\left (\int \frac {1+c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right ) \\ & = -\frac {3 b c}{10 x^2}-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} b c^{5/3} \log \left (1+c^{2/3} x^2\right )-\frac {1}{20} \left (b c^{5/3}\right ) \text {Subst}\left (\int \frac {-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {1}{20} \left (3 b c^{7/3}\right ) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right ) \\ & = -\frac {3 b c}{10 x^2}-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} b c^{5/3} \log \left (1+c^{2/3} x^2\right )-\frac {1}{20} b c^{5/3} \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )-\frac {1}{10} \left (3 b c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right ) \\ & = -\frac {3 b c}{10 x^2}-\frac {a+b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} \sqrt {3} b c^{5/3} \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )+\frac {1}{10} b c^{5/3} \log \left (1+c^{2/3} x^2\right )-\frac {1}{20} b c^{5/3} \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.59 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {3 b c}{10 x^2}-\frac {b \arctan \left (c x^3\right )}{5 x^5}+\frac {1}{10} \sqrt {3} b c^{5/3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{10} \sqrt {3} b c^{5/3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )+\frac {1}{10} b c^{5/3} \log \left (1+c^{2/3} x^2\right )-\frac {1}{20} b c^{5/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-\frac {1}{20} b c^{5/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right ) \]
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Time = 0.76 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {a}{5 x^{5}}-\frac {b \arctan \left (c \,x^{3}\right )}{5 x^{5}}-\frac {3 b c}{10 x^{2}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{10 \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{4}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{20 \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{10 \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}\) | \(105\) |
parts | \(-\frac {a}{5 x^{5}}-\frac {b \arctan \left (c \,x^{3}\right )}{5 x^{5}}-\frac {3 b c}{10 x^{2}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{10 \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{4}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{20 \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{10 \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}\) | \(105\) |
risch | \(\frac {i b \ln \left (i c \,x^{3}+1\right )}{10 x^{5}}-\frac {i b \ln \left (-i c \,x^{3}+1\right )}{10 x^{5}}-\frac {3 b c}{10 x^{2}}-\frac {b c \ln \left (x +\left (\frac {i}{c}\right )^{\frac {1}{3}}\right )}{10 \left (\frac {i}{c}\right )^{\frac {2}{3}}}+\frac {b c \ln \left (x^{2}-\left (\frac {i}{c}\right )^{\frac {1}{3}} x +\left (\frac {i}{c}\right )^{\frac {2}{3}}\right )}{20 \left (\frac {i}{c}\right )^{\frac {2}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {i}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{10 \left (\frac {i}{c}\right )^{\frac {2}{3}}}-\frac {a}{5 x^{5}}-\frac {b c \ln \left (x -\left (\frac {i}{c}\right )^{\frac {1}{3}}\right )}{10 \left (\frac {i}{c}\right )^{\frac {2}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {i}{c}\right )^{\frac {1}{3}} x +\left (\frac {i}{c}\right )^{\frac {2}{3}}\right )}{20 \left (\frac {i}{c}\right )^{\frac {2}{3}}}+\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {i}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{10 \left (\frac {i}{c}\right )^{\frac {2}{3}}}\) | \(236\) |
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=-\frac {2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{3}} c x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} - \frac {1}{3} \, \sqrt {3}\right ) + b {\left (c^{2}\right )}^{\frac {1}{3}} c x^{5} \log \left (c^{2} x^{4} - {\left (c^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {1}{3}} c x^{5} \log \left (c^{2} x^{2} + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, b c x^{3} + 4 \, b \arctan \left (c x^{3}\right ) + 4 \, a}{20 \, x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (112) = 224\).
Time = 61.14 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.49 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=\begin {cases} - \frac {a}{5 x^{5}} + \frac {b c^{2} \sqrt [6]{- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{5} - \frac {b c \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{5 \sqrt [3]{- \frac {1}{c^{2}}}} + \frac {3 b c \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{20 \sqrt [3]{- \frac {1}{c^{2}}}} - \frac {b c \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{20 \sqrt [3]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3} b c \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{10 \sqrt [3]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3} b c \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{10 \sqrt [3]{- \frac {1}{c^{2}}}} - \frac {3 b c}{10 x^{2}} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{5 x^{5}} & \text {for}\: c \neq 0 \\- \frac {a}{5 x^{5}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=-\frac {1}{20} \, {\left ({\left (2 \, \sqrt {3} c^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right ) + c^{\frac {2}{3}} \log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right ) - 2 \, c^{\frac {2}{3}} \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right ) + \frac {6}{x^{2}}\right )} c + \frac {4 \, \arctan \left (c x^{3}\right )}{x^{5}}\right )} b - \frac {a}{5 \, x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=-\frac {1}{20} \, b c^{3} {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{2}} + \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{4} - \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{2}} - \frac {2 \, \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {4}{3}}}\right )} - \frac {3 \, b c x^{3} + 2 \, b \arctan \left (c x^{3}\right ) + 2 \, a}{10 \, x^{5}} \]
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Time = 2.71 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^6} \, dx=\frac {b\,c^{5/3}\,\ln \left (c^{2/3}\,x^2+1\right )}{10}-\frac {\frac {3\,b\,c\,x^3}{2}+a}{5\,x^5}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{5\,x^5}-\frac {b\,c^{5/3}\,\ln \left (\sqrt {3}\,c^{2/3}\,x^2-c^{2/3}\,x^2\,1{}\mathrm {i}+2{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{20}+\frac {b\,c^{5/3}\,\ln \left (-c^{2/3}\,x^2\,1{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x^2+2{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{20} \]
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