Optimal. Leaf size=101 \[ a x+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6091, 275, 292, 31, 634, 617, 204, 628} \[ a x+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 617
Rule 628
Rule 634
Rule 6091
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c x^3\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )-(3 b c) \int \frac {x^3}{1-c^2 x^6} \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )-\frac {1}{2} (3 b c) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x^3} \, dx,x,x^2\right )\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )-\frac {1}{2} \left (b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x} \, dx,x,x^2\right )+\frac {1}{2} \left (b \sqrt [3]{c}\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 )\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \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 )}{4 \sqrt [3]{c}}+\frac {1}{4} \left (3 b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=a x+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 136, normalized size = 1.35 \[ a x-\frac {b \left (\log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-2 \log \left (1-\sqrt [3]{c} x\right )-2 \log \left (\sqrt [3]{c} x+1\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x-1}{\sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )\right )}{4 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 260, normalized size = 2.57 \[ \left [\frac {\sqrt {3} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (\frac {2 \, c^{2} x^{6} - 3 \, c^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (2 \, c^{\frac {5}{3}} x^{4} - c x^{2} - c^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} + 1}{c^{2} x^{6} - 1}\right ) + 2 \, b c x \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} + c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} - c^{\frac {1}{3}}\right )}{4 \, c}, \frac {2 \, b c x \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, \sqrt {3} b c^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c x^{2} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} + c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} - c^{\frac {1}{3}}\right )}{4 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 109, normalized size = 1.08 \[ \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{2}} - \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{4} + \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{2}} + \frac {2 \, \log \left ({\left | x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {4}{3}}}\right )} + 2 \, x \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 99, normalized size = 0.98 \[ a x +b x \arctanh \left (c \,x^{3}\right )+\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 90, normalized size = 0.89 \[ \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} - \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} + 4 \, x \operatorname {artanh}\left (c x^{3}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.76, size = 107, normalized size = 1.06 \[ a\,x+\frac {b\,\ln \left (c^{2/3}\,x^2-1\right )}{2\,c^{1/3}}-\frac {\ln \left (4\,c^{2/3}\,x^2+2-\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}}-\frac {\ln \left (4\,c^{2/3}\,x^2+2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}}+\frac {b\,x\,\ln \left (c\,x^3+1\right )}{2}-\frac {b\,x\,\ln \left (1-c\,x^3\right )}{2} \]
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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