Optimal. Leaf size=104 \[ -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \tan ^{-1}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6097, 275, 200, 31, 634, 617, 204, 628} \[ -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )+\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \tan ^{-1}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 275
Rule 617
Rule 628
Rule 634
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{x^2} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}+(3 b c) \int \frac {x}{1-c^2 x^6} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}+\frac {1}{2} (3 b c) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^3} \, dx,x,x^2\right )\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x} \, dx,x,x^2\right )+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {2+c^{2/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac {1}{4} \left (b \sqrt [3]{c}\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}{4} (3 b c) \operatorname {Subst}\left (\int \frac {1}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )-\frac {1}{2} \left (3 b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 c^{2/3} x^2\right )\\ &=\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \tan ^{-1}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 183, normalized size = 1.76 \[ -\frac {a}{x}+\frac {1}{4} b \sqrt [3]{c} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-\frac {b \tanh ^{-1}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-\sqrt [3]{c} x\right )-\frac {1}{2} b \sqrt [3]{c} \log \left (\sqrt [3]{c} x+1\right )+\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x-1}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 117, normalized size = 1.12 \[ -\frac {2 \, \sqrt {3} b \left (-c\right )^{\frac {1}{3}} x \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c\right )^{\frac {2}{3}} x^{2} + \frac {1}{3} \, \sqrt {3}\right ) + b \left (-c\right )^{\frac {1}{3}} x \log \left (c^{2} x^{4} - \left (-c\right )^{\frac {1}{3}} c x^{2} + \left (-c\right )^{\frac {2}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {1}{3}} x \log \left (c x^{2} + \left (-c\right )^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{4 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 106, normalized size = 1.02 \[ \frac {1}{4} \, b c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{{\left | c \right |}^{\frac {2}{3}}} + \frac {\log \left (x^{4} + \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{{\left | c \right |}^{\frac {2}{3}}} - \frac {2 \, \log \left ({\left | x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {2}{3}}}\right )} - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{2 \, x} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 105, normalized size = 1.01 \[ -\frac {a}{x}-\frac {b \arctanh \left (c \,x^{3}\right )}{x}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 94, normalized size = 0.90 \[ \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {2}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}}\right )} - \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x}\right )} b - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 117, normalized size = 1.12 \[ \frac {b\,\ln \left (1-c\,x^3\right )}{2\,x}-\frac {b\,c^{1/3}\,\ln \left (1-c^{2/3}\,x^2\right )}{2}-\frac {b\,\ln \left (c\,x^3+1\right )}{2\,x}-\frac {a}{x}-\frac {b\,c^{1/3}\,\ln \left (-\sqrt {3}-c^{2/3}\,x^2\,2{}\mathrm {i}-\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}+\frac {b\,c^{1/3}\,\ln \left (-\sqrt {3}+c^{2/3}\,x^2\,2{}\mathrm {i}+1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4} \]
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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