Optimal. Leaf size=174 \[ -\frac {3 b c}{4 x}+\frac {1}{8} \sqrt {3} b c^{4/3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {3} b c^{4/3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} b c^{4/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} b c^{4/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6037, 331,
302, 648, 632, 210, 642, 212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}+\frac {1}{8} \sqrt {3} b c^{4/3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {3} b c^{4/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )-\frac {1}{16} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac {1}{16} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )+\frac {1}{4} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {3 b c}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 302
Rule 331
Rule 632
Rule 642
Rule 648
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{x^5} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}+\frac {1}{4} (3 b c) \int \frac {1}{x^2 \left (1-c^2 x^6\right )} \, dx\\ &=-\frac {3 b c}{4 x}-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}+\frac {1}{4} \left (3 b c^3\right ) \int \frac {x^4}{1-c^2 x^6} \, dx\\ &=-\frac {3 b c}{4 x}-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}+\frac {1}{4} \left (b c^{5/3}\right ) \int \frac {1}{1-c^{2/3} x^2} \, dx+\frac {1}{4} \left (b c^{5/3}\right ) \int \frac {-\frac {1}{2}-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{4} \left (b c^{5/3}\right ) \int \frac {-\frac {1}{2}+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=-\frac {3 b c}{4 x}+\frac {1}{4} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} \left (b c^{4/3}\right ) \int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{16} \left (b c^{4/3}\right ) \int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx-\frac {1}{16} \left (3 b c^{5/3}\right ) \int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx-\frac {1}{16} \left (3 b c^{5/3}\right ) \int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=-\frac {3 b c}{4 x}+\frac {1}{4} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} b c^{4/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} b c^{4/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )-\frac {1}{8} \left (3 b c^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )+\frac {1}{8} \left (3 b c^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )\\ &=-\frac {3 b c}{4 x}+\frac {1}{8} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} b c^{4/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} b c^{4/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 196, normalized size = 1.13 \begin {gather*} -\frac {a}{4 x^4}-\frac {3 b c}{4 x}-\frac {1}{8} \sqrt {3} b c^{4/3} \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {3} b c^{4/3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {b \tanh ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} x\right )+\frac {1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} x\right )-\frac {1}{16} b c^{4/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} b c^{4/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 172, normalized size = 0.99
method | result | size |
default | \(-\frac {a}{4 x^{4}}-\frac {b \arctanh \left (c \,x^{3}\right )}{4 x^{4}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {3 b c}{4 x}\) | \(172\) |
risch | \(-\frac {b \ln \left (c \,x^{3}+1\right )}{8 x^{4}}-\frac {a}{4 x^{4}}+\frac {b \ln \left (-c \,x^{3}+1\right )}{8 x^{4}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {3 b c}{4 x}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(189\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 160, normalized size = 0.92 \begin {gather*} -\frac {1}{16} \, {\left ({\left (2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) - c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right ) + c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right ) - 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right ) + \frac {12}{x}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 196, normalized size = 1.13 \begin {gather*} -\frac {2 \, \sqrt {3} b \left (-c\right )^{\frac {1}{3}} c x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c\right )^{\frac {1}{3}} x - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {3} b c^{\frac {4}{3}} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} c^{\frac {1}{3}} x - \frac {1}{3} \, \sqrt {3}\right ) + b \left (-c\right )^{\frac {1}{3}} c x^{4} \log \left (c x^{2} + \left (-c\right )^{\frac {2}{3}} x - \left (-c\right )^{\frac {1}{3}}\right ) + b c^{\frac {4}{3}} x^{4} \log \left (c x^{2} - c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {1}{3}} c x^{4} \log \left (c x - \left (-c\right )^{\frac {2}{3}}\right ) - 2 \, b c^{\frac {4}{3}} x^{4} \log \left (c x + c^{\frac {2}{3}}\right ) + 12 \, b c x^{3} + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{16 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 187, normalized size = 1.07 \begin {gather*} -\frac {1}{8} \, \sqrt {3} b c {\left | c \right |}^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right ) - \frac {1}{8} \, \sqrt {3} b c {\left | c \right |}^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right ) + \frac {b c^{3} \log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{16 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c^{3} \log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{16 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {1}{8} \, b c {\left | c \right |}^{\frac {1}{3}} \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right ) - \frac {b c^{3} \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{8 \, x^{4}} - \frac {3 \, b c x^{3} + a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 125, normalized size = 0.72 \begin {gather*} \frac {b\,\ln \left (1-c\,x^3\right )}{8\,x^4}-\frac {b\,c^{4/3}\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{4}-\frac {3\,b\,c}{4\,x}-\frac {b\,\ln \left (c\,x^3+1\right )}{8\,x^4}-\frac {a}{4\,x^4}-\frac {\sqrt {3}\,b\,c^{4/3}\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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