Optimal. Leaf size=54 \[ \frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac{b x^2}{8 c^3}-\frac{b \tan ^{-1}\left (c x^2\right )}{8 c^4}-\frac{b x^6}{24 c} \]
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Rubi [A] time = 0.0351121, antiderivative size = 54, normalized size of antiderivative = 1., 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 5033
Rule 275
Rule 302
Rule 203
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*}
Mathematica [A] time = 0.0080472, size = 59, normalized size = 1.09 \[ \frac{a x^8}{8}+\frac{b x^2}{8 c^3}-\frac{b \tan ^{-1}\left (c x^2\right )}{8 c^4}-\frac{b x^6}{24 c}+\frac{1}{8} b x^8 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 50, normalized size = 0.9 \begin{align*}{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45766, size = 73, normalized size = 1.35 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62789, size = 111, normalized size = 2.06 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 128.876, size = 58, normalized size = 1.07 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09653, size = 81, normalized size = 1.5 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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