Optimal. Leaf size=54 \[ \frac {1}{8} b c^3 x^2+\frac {1}{24} b c x^6+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )-\frac {1}{8} b c^4 \tanh ^{-1}\left (\frac {x^2}{c}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6037, 269, 281,
308, 213} \begin {gather*} \frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )-\frac {1}{8} b c^4 \tanh ^{-1}\left (\frac {x^2}{c}\right )+\frac {1}{8} b c^3 x^2+\frac {1}{24} b c x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 269
Rule 281
Rule 308
Rule 6037
Rubi steps
\begin {align*} \int x^7 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right ) \, dx &=\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )+\frac {1}{4} (b c) \int \frac {x^5}{1-\frac {c^2}{x^4}} \, dx\\ &=\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )+\frac {1}{4} (b c) \int \frac {x^9}{-c^2+x^4} \, dx\\ &=\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )+\frac {1}{8} (b c) \text {Subst}\left (\int \frac {x^4}{-c^2+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )+\frac {1}{8} (b c) \text {Subst}\left (\int \left (c^2+x^2+\frac {c^4}{-c^2+x^2}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{8} b c^3 x^2+\frac {1}{24} b c x^6+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )+\frac {1}{8} \left (b c^5\right ) \text {Subst}\left (\int \frac {1}{-c^2+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{8} b c^3 x^2+\frac {1}{24} b c x^6+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )-\frac {1}{8} b c^4 \tanh ^{-1}\left (\frac {x^2}{c}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 73, normalized size = 1.35 \begin {gather*} \frac {1}{8} b c^3 x^2+\frac {1}{24} b c x^6+\frac {a x^8}{8}+\frac {1}{8} b x^8 \tanh ^{-1}\left (\frac {c}{x^2}\right )+\frac {1}{16} b c^4 \log \left (-c+x^2\right )-\frac {1}{16} b c^4 \log \left (c+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 64, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {x^{8} a}{8}+\frac {b \,x^{8} \arctanh \left (\frac {c}{x^{2}}\right )}{8}+\frac {b c \,x^{6}}{24}+\frac {b \,c^{3} x^{2}}{8}-\frac {b \,c^{4} \ln \left (1+\frac {c}{x^{2}}\right )}{16}+\frac {b \,c^{4} \ln \left (\frac {c}{x^{2}}-1\right )}{16}\) | \(64\) |
default | \(\frac {x^{8} a}{8}+\frac {b \,x^{8} \arctanh \left (\frac {c}{x^{2}}\right )}{8}+\frac {b c \,x^{6}}{24}+\frac {b \,c^{3} x^{2}}{8}-\frac {b \,c^{4} \ln \left (1+\frac {c}{x^{2}}\right )}{16}+\frac {b \,c^{4} \ln \left (\frac {c}{x^{2}}-1\right )}{16}\) | \(64\) |
risch | \(\frac {x^{8} b \ln \left (x^{2}+c \right )}{16}-\frac {x^{8} b \ln \left (-x^{2}+c \right )}{16}+\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}}{32}-\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}{32}-\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}}{32}+\frac {i \pi b \,x^{8} \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}}{32}-\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{3}}{32}-\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{3}}{32}+\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}}{16}+\frac {i \pi b \,x^{8} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}{32}-\frac {i \pi b \,x^{8}}{16}-\frac {i \pi b \,x^{8} \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}}{32}+\frac {x^{8} a}{8}+\frac {b c \,x^{6}}{24}+\frac {b \,c^{3} x^{2}}{8}-\frac {b \,c^{4} \ln \left (-x^{2}-c \right )}{16}+\frac {b \,c^{4} \ln \left (-x^{2}+c \right )}{16}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 62, normalized size = 1.15 \begin {gather*} \frac {1}{8} \, a x^{8} + \frac {1}{48} \, {\left (6 \, x^{8} \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + {\left (2 \, x^{6} + 6 \, c^{2} x^{2} - 3 \, c^{3} \log \left (x^{2} + c\right ) + 3 \, c^{3} \log \left (x^{2} - c\right )\right )} c\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 53, normalized size = 0.98 \begin {gather*} \frac {1}{8} \, a x^{8} + \frac {1}{24} \, b c x^{6} + \frac {1}{8} \, b c^{3} x^{2} + \frac {1}{16} \, {\left (b x^{8} - b c^{4}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.10, size = 51, normalized size = 0.94 \begin {gather*} \frac {a x^{8}}{8} - \frac {b c^{4} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{8} + \frac {b c^{3} x^{2}}{8} + \frac {b c x^{6}}{24} + \frac {b x^{8} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 71, normalized size = 1.31 \begin {gather*} \frac {1}{16} \, b x^{8} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + \frac {1}{8} \, a x^{8} + \frac {1}{24} \, b c x^{6} + \frac {1}{8} \, b c^{3} x^{2} - \frac {1}{16} \, b c^{4} \log \left (x^{2} + c\right ) + \frac {1}{16} \, b c^{4} \log \left (-x^{2} + c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.01, size = 66, normalized size = 1.22 \begin {gather*} \frac {a\,x^8}{8}+\frac {b\,c^3\,x^2}{8}+\frac {b\,x^8\,\ln \left (x^2+c\right )}{16}+\frac {b\,c\,x^6}{24}-\frac {b\,x^8\,\ln \left (x^2-c\right )}{16}+\frac {b\,c^4\,\mathrm {atan}\left (\frac {x^2\,1{}\mathrm {i}}{c}\right )\,1{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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