Optimal. Leaf size=63 \[ -\frac {2 b c}{15 x^3}+\frac {1}{5} b c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right )+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{5 x^5} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6037, 331, 218,
212, 209} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} b c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right )+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {2 b c}{15 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 331
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{x^6} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} (2 b c) \int \frac {1}{x^4 \left (1-c^2 x^4\right )} \, dx\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} \left (2 b c^3\right ) \int \frac {1}{1-c^2 x^4} \, dx\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} \left (b c^3\right ) \int \frac {1}{1-c x^2} \, dx+\frac {1}{5} \left (b c^3\right ) \int \frac {1}{1+c x^2} \, dx\\ &=-\frac {2 b c}{15 x^3}+\frac {1}{5} b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{5 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 91, normalized size = 1.44 \begin {gather*} -\frac {a}{5 x^5}-\frac {2 b c}{15 x^3}+\frac {1}{5} b c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right )-\frac {b \tanh ^{-1}\left (c x^2\right )}{5 x^5}-\frac {1}{10} b c^{5/2} \log \left (1-\sqrt {c} x\right )+\frac {1}{10} b c^{5/2} \log \left (1+\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 51, normalized size = 0.81
method | result | size |
default | \(-\frac {a}{5 x^{5}}-\frac {b \arctanh \left (c \,x^{2}\right )}{5 x^{5}}+\frac {b \,c^{\frac {5}{2}} \arctanh \left (x \sqrt {c}\right )}{5}+\frac {b \,c^{\frac {5}{2}} \arctan \left (x \sqrt {c}\right )}{5}-\frac {2 b c}{15 x^{3}}\) | \(51\) |
risch | \(-\frac {b \ln \left (c \,x^{2}+1\right )}{10 x^{5}}-\frac {-3 c^{\frac {5}{2}} b \ln \left (c^{\frac {19}{2}}+x \,c^{10}\right ) x^{5}+3 c^{\frac {5}{2}} b \ln \left (-c^{\frac {19}{2}}+x \,c^{10}\right ) x^{5}-3 c^{2} \sqrt {-c}\, b \ln \left (c^{2} \sqrt {-c}+x \,c^{3}\right ) x^{5}+3 c^{2} \sqrt {-c}\, b \ln \left (-c^{2} \sqrt {-c}+x \,c^{3}\right ) x^{5}+4 b c \,x^{2}-3 b \ln \left (-c \,x^{2}+1\right )+6 a}{30 x^{5}}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 66, normalized size = 1.05 \begin {gather*} \frac {1}{30} \, {\left ({\left (6 \, c^{\frac {3}{2}} \arctan \left (\sqrt {c} x\right ) - 3 \, c^{\frac {3}{2}} \log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right ) - \frac {4}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x^{2}\right )}{x^{5}}\right )} b - \frac {a}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (47) = 94\).
time = 0.36, size = 187, normalized size = 2.97 \begin {gather*} \left [\frac {6 \, b c^{\frac {5}{2}} x^{5} \arctan \left (\sqrt {c} x\right ) + 3 \, b c^{\frac {5}{2}} x^{5} \log \left (\frac {c x^{2} + 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right ) - 4 \, b c x^{2} - 3 \, b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) - 6 \, a}{30 \, x^{5}}, -\frac {6 \, b \sqrt {-c} c^{2} x^{5} \arctan \left (\sqrt {-c} x\right ) - 3 \, b \sqrt {-c} c^{2} x^{5} \log \left (\frac {c x^{2} + 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right ) + 4 \, b c x^{2} + 3 \, b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 6 \, a}{30 \, x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1948 vs.
\(2 (61) = 122\).
time = 8.69, size = 1948, normalized size = 30.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 91, normalized size = 1.44 \begin {gather*} \frac {1}{10} \, b c^{3} {\left (\frac {2 \, \arctan \left (x \sqrt {{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} + \frac {\log \left ({\left | x + \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{\sqrt {{\left | c \right |}}} - \frac {\log \left ({\left | x - \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{\sqrt {{\left | c \right |}}}\right )} - \frac {b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{10 \, x^{5}} - \frac {2 \, b c x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 71, normalized size = 1.13 \begin {gather*} \frac {b\,c^{5/2}\,\mathrm {atan}\left (\sqrt {c}\,x\right )}{5}-\frac {\frac {2\,b\,c\,x^2}{3}+a}{5\,x^5}-\frac {b\,\ln \left (c\,x^2+1\right )}{10\,x^5}+\frac {b\,\ln \left (1-c\,x^2\right )}{10\,x^5}-\frac {b\,c^{5/2}\,\mathrm {atan}\left (\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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