Optimal. Leaf size=115 \[ -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{5 x^5}-\frac {1}{10} b c^{5/3} \log \left (1-c^{2/3} x^2\right )-\frac {1}{10} \sqrt {3} b c^{5/3} \tan ^{-1}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )+\frac {1}{20} b c^{5/3} \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )-\frac {3 b c}{10 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6097, 275, 325, 292, 31, 634, 617, 204, 628} \[ -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{5 x^5}-\frac {1}{10} b c^{5/3} \log \left (1-c^{2/3} x^2\right )+\frac {1}{20} b c^{5/3} \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )-\frac {1}{10} \sqrt {3} b c^{5/3} \tan ^{-1}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )-\frac {3 b c}{10 x^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 325
Rule 617
Rule 628
Rule 634
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{x^6} \, dx &=-\frac {a+b \tanh ^{-1}\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 \tanh ^{-1}\left (c x^3\right )}{5 x^5}+\frac {1}{10} (3 b c) \operatorname {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 \tanh ^{-1}\left (c x^3\right )}{5 x^5}+\frac {1}{10} \left (3 b c^3\right ) \operatorname {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 \tanh ^{-1}\left (c x^3\right )}{5 x^5}+\frac {1}{10} \left (b c^{7/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x} \, dx,x,x^2\right )-\frac {1}{10} \left (b c^{7/3}\right ) \operatorname {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 \tanh ^{-1}\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 ) \operatorname {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 ) \operatorname {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 \tanh ^{-1}\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 ) \operatorname {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 {1}{10} \sqrt {3} b c^{5/3} \tan ^{-1}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )-\frac {a+b \tanh ^{-1}\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 )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 196, normalized size = 1.70 \[ -\frac {a}{5 x^5}+\frac {1}{20} b c^{5/3} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac {1}{20} b c^{5/3} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-\frac {1}{10} b c^{5/3} \log \left (1-\sqrt [3]{c} x\right )-\frac {1}{10} b c^{5/3} \log \left (\sqrt [3]{c} x+1\right )-\frac {1}{10} \sqrt {3} b c^{5/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x-1}{\sqrt {3}}\right )+\frac {1}{10} \sqrt {3} b c^{5/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )-\frac {3 b c}{10 x^2}-\frac {b \tanh ^{-1}\left (c x^3\right )}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 151, normalized size = 1.31 \[ -\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} b c x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} x^{2} - \frac {1}{3} \, \sqrt {3}\right ) + \left (-c^{2}\right )^{\frac {1}{3}} b c x^{5} \log \left (c^{2} x^{4} + \left (-c^{2}\right )^{\frac {2}{3}} x^{2} - \left (-c^{2}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {1}{3}} b c x^{5} \log \left (c^{2} x^{2} - \left (-c^{2}\right )^{\frac {2}{3}}\right ) + 6 \, b c x^{3} + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{20 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 125, normalized size = 1.09 \[ -\frac {1}{20} \, b c^{3} {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{2}} - \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{4} + \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{2}} + \frac {2 \, \log \left ({\left | x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {4}{3}}}\right )} - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{10 \, x^{5}} - \frac {3 \, b c x^{3} + 2 \, a}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 172, normalized size = 1.50 \[ -\frac {a}{5 x^{5}}-\frac {b \arctanh \left (c \,x^{3}\right )}{5 x^{5}}-\frac {3 b c}{10 x^{2}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{10 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{20 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{10 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{10 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b c \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{20 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{10 \left (\frac {1}{c}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 100, normalized size = 0.87 \[ -\frac {1}{20} \, {\left ({\left (2 \, \sqrt {3} c^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right ) - c^{\frac {2}{3}} \log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right ) + 2 \, c^{\frac {2}{3}} \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right ) + \frac {6}{x^{2}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{5}}\right )} b - \frac {a}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.12, size = 135, normalized size = 1.17 \[ \frac {b\,\ln \left (1-c\,x^3\right )}{10\,x^5}-\frac {b\,c^{5/3}\,\ln \left (c^{2/3}\,x^2-1\right )}{10}-\frac {b\,\ln \left (c\,x^3+1\right )}{10\,x^5}-\frac {\frac {3\,b\,c\,x^3}{2}+a}{5\,x^5}+\frac {b\,c^{5/3}\,\ln \left (\sqrt {3}\,c^{2/3}\,x^2-c^{2/3}\,x^2\,1{}\mathrm {i}-2{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{20}-\frac {b\,c^{5/3}\,\ln \left (-c^{2/3}\,x^2\,1{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x^2-2{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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