Optimal. Leaf size=176 \[ \frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {b \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{16 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {3 b x^5}{40 c} \]
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Rubi [A] time = 0.28, antiderivative size = 176, 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 = {6097, 321, 296, 634, 618, 204, 628, 206} \[ \frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {b \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{16 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {3 b x^5}{40 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 296
Rule 321
Rule 618
Rule 628
Rule 634
Rule 6097
Rubi steps
\begin {align*} \int x^7 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {1}{8} (3 b c) \int \frac {x^{10}}{1-c^2 x^6} \, dx\\ &=\frac {3 b x^5}{40 c}+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {(3 b) \int \frac {x^4}{1-c^2 x^6} \, dx}{8 c}\\ &=\frac {3 b x^5}{40 c}+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac {b \int \frac {1}{1-c^{2/3} x^2} \, dx}{8 c^{7/3}}-\frac {b \int \frac {-\frac {1}{2}-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{7/3}}-\frac {b \int \frac {-\frac {1}{2}+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{7/3}}\\ &=\frac {3 b x^5}{40 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{8/3}}-\frac {b \int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{8/3}}+\frac {(3 b) \int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{7/3}}+\frac {(3 b) \int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{7/3}}\\ &=\frac {3 b x^5}{40 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )}{16 c^{8/3}}\\ &=\frac {3 b x^5}{40 c}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{16 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {1}{8} x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 198, normalized size = 1.12 \[ \frac {a x^8}{8}+\frac {b \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {b \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac {b \log \left (1-\sqrt [3]{c} x\right )}{16 c^{8/3}}-\frac {b \log \left (\sqrt [3]{c} x+1\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x-1}{\sqrt {3}}\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{16 c^{8/3}}+\frac {3 b x^5}{40 c}+\frac {1}{8} b x^8 \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 248, normalized size = 1.41 \[ \frac {10 \, b c^{4} x^{8} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 20 \, a c^{4} x^{8} + 12 \, b c^{3} x^{5} + 10 \, \sqrt {3} b c \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c x + \left (-c^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}}}{3 \, c}\right ) + 10 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} {\left (c^{2}\right )}^{\frac {1}{6}} {\left (2 \, c x + {\left (c^{2}\right )}^{\frac {1}{3}}\right )}}{3 \, c}\right ) + 5 \, \left (-c^{2}\right )^{\frac {2}{3}} b \log \left (c^{2} x^{2} + \left (-c^{2}\right )^{\frac {1}{3}} c x + \left (-c^{2}\right )^{\frac {2}{3}}\right ) - 5 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c^{2} x^{2} + {\left (c^{2}\right )}^{\frac {1}{3}} c x + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 10 \, \left (-c^{2}\right )^{\frac {2}{3}} b \log \left (c x - \left (-c^{2}\right )^{\frac {1}{3}}\right ) + 10 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c x - {\left (c^{2}\right )}^{\frac {1}{3}}\right )}{160 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 208, normalized size = 1.18 \[ \frac {1}{16} \, b x^{8} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + \frac {1}{8} \, a x^{8} + \frac {3 \, b x^{5}}{40 \, c} - \frac {b \left (-\frac {1}{c}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {1}{c}\right )^{\frac {1}{3}} \right |}\right )}{16 \, c^{2}} + \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {1}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{16 \, \left (-c^{2}\right )^{\frac {1}{3}} c^{2}} + \frac {\sqrt {3} b {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} c^{\frac {1}{3}} {\left (2 \, x + \frac {1}{c^{\frac {1}{3}}}\right )}\right )}{16 \, c^{4}} - \frac {b \log \left (x^{2} + x \left (-\frac {1}{c}\right )^{\frac {1}{3}} + \left (-\frac {1}{c}\right )^{\frac {2}{3}}\right )}{32 \, \left (-c^{2}\right )^{\frac {1}{3}} c^{2}} - \frac {b \log \left (x^{2} + \frac {x}{c^{\frac {1}{3}}} + \frac {1}{c^{\frac {2}{3}}}\right )}{32 \, c^{2} {\left | c \right |}^{\frac {2}{3}}} + \frac {b \log \left ({\left | x - \frac {1}{c^{\frac {1}{3}}} \right |}\right )}{16 \, c^{\frac {8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 186, normalized size = 1.06 \[ \frac {x^{8} a}{8}+\frac {b \,x^{8} \arctanh \left (c \,x^{3}\right )}{8}+\frac {3 b \,x^{5}}{40 c}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{32 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{32 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 164, normalized size = 0.93 \[ \frac {1}{8} \, a x^{8} + \frac {1}{160} \, {\left (20 \, x^{8} \operatorname {artanh}\left (c x^{3}\right ) + {\left (\frac {12 \, x^{5}}{c^{2}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} - \frac {5 \, \log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {11}{3}}} + \frac {5 \, \log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {11}{3}}} - \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}}\right )} c\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 127, normalized size = 0.72 \[ \frac {a\,x^8}{8}+\frac {b\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{8\,c^{8/3}}+\frac {3\,b\,x^5}{40\,c}+\frac {b\,x^8\,\ln \left (c\,x^3+1\right )}{16}-\frac {b\,x^8\,\ln \left (1-c\,x^3\right )}{16}+\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{16\,c^{8/3}} \]
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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