Optimal. Leaf size=54 \[ \frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \tan ^{-1}\left (c x^2\right )}{8 c^4}+\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c} \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5033, 275, 302, 203} \[ \frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac {b x^2}{8 c^3}-\frac {b \tan ^{-1}\left (c x^2\right )}{8 c^4}-\frac {b x^6}{24 c} \]
Antiderivative was successfully verified.
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Rule 203
Rule 275
Rule 302
Rule 5033
Rubi steps
\begin {align*} \int x^7 \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{4} (b c) \int \frac {x^9}{1+c^2 x^4} \, dx\\ &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{8} (b c) \operatorname {Subst}\left (\int \frac {x^4}{1+c^2 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{8} (b c) \operatorname {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^2\right )}{8 c^3}\\ &=\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}-\frac {b \tan ^{-1}\left (c x^2\right )}{8 c^4}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 1.09 \[ \frac {a x^8}{8}-\frac {b \tan ^{-1}\left (c x^2\right )}{8 c^4}+\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}+\frac {1}{8} b x^8 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 51, normalized size = 0.94 \[ \frac {3 \, a c^{4} x^{8} - b c^{3} x^{6} + 3 \, b c x^{2} + 3 \, {\left (b c^{4} x^{8} - b\right )} \arctan \left (c x^{2}\right )}{24 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 1.11 \[ \frac {3 \, a c x^{8} + {\left (3 \, c x^{8} \arctan \left (c x^{2}\right ) - \frac {3 \, \arctan \left (c x^{2}\right )}{c^{3}} - \frac {c^{9} x^{6} - 3 \, c^{7} x^{2}}{c^{9}}\right )} b}{24 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 50, normalized size = 0.93 \[ \frac {x^{8} a}{8}+\frac {b \,x^{8} \arctan \left (c \,x^{2}\right )}{8}-\frac {b \,x^{6}}{24 c}+\frac {b \,x^{2}}{8 c^{3}}-\frac {b \arctan \left (c \,x^{2}\right )}{8 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 54, normalized size = 1.00 \[ \frac {1}{8} \, a x^{8} + \frac {1}{24} \, {\left (3 \, x^{8} \arctan \left (c x^{2}\right ) - c {\left (\frac {c^{2} x^{6} - 3 \, x^{2}}{c^{4}} + \frac {3 \, \arctan \left (c x^{2}\right )}{c^{5}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 49, normalized size = 0.91 \[ \frac {a\,x^8}{8}+\frac {b\,x^2}{8\,c^3}-\frac {b\,x^6}{24\,c}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{8\,c^4}+\frac {b\,x^8\,\mathrm {atan}\left (c\,x^2\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 65.10, size = 58, normalized size = 1.07 \[ \begin {cases} \frac {a x^{8}}{8} + \frac {b x^{8} \operatorname {atan}{\left (c x^{2} \right )}}{8} - \frac {b x^{6}}{24 c} + \frac {b x^{2}}{8 c^{3}} - \frac {b \operatorname {atan}{\left (c x^{2} \right )}}{8 c^{4}} & \text {for}\: c \neq 0 \\\frac {a x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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