Optimal. Leaf size=89 \[ \frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{20 c}-\frac{3 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{40 c^3}-\frac{3 b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{40 c^5} \]
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Rubi [A] time = 0.0463687, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5220, 266, 51, 63, 208} \[ \frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{20 c}-\frac{3 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{40 c^3}-\frac{3 b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{40 c^5} \]
Antiderivative was successfully verified.
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Rule 5220
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^4 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \int \frac{x^3}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{5 c}\\ &=\frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{10 c}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{40 c^3}\\ &=-\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x^2}{40 c^3}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{80 c^5}\\ &=-\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x^2}{40 c^3}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{c^2-c^2 x^2} \, dx,x,\sqrt{1-\frac{1}{c^2 x^2}}\right )}{40 c^3}\\ &=-\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x^2}{40 c^3}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac{3 b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{40 c^5}\\ \end{align*}
Mathematica [A] time = 0.0688045, size = 97, normalized size = 1.09 \[ \frac{a x^5}{5}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left (-\frac{3 x^2}{40 c^3}-\frac{x^4}{20 c}\right )-\frac{3 b \log \left (x \left (\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{40 c^5}+\frac{1}{5} b x^5 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 150, normalized size = 1.7 \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{{x}^{5}b{\rm arcsec} \left (cx\right )}{5}}-{\frac{b{x}^{4}}{20\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{x}^{2}}{40\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,b}{40\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{3\,b}{40\,{c}^{6}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00718, size = 177, normalized size = 1.99 \begin{align*} \frac{1}{5} \, a x^{5} + \frac{1}{80} \,{\left (16 \, x^{5} \operatorname{arcsec}\left (c x\right ) + \frac{\frac{2 \,{\left (3 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.79015, size = 248, normalized size = 2.79 \begin{align*} \frac{8 \, a c^{5} x^{5} + 16 \, b c^{5} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 8 \,{\left (b c^{5} x^{5} - b c^{5}\right )} \operatorname{arcsec}\left (c x\right ) + 3 \, b \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt{c^{2} x^{2} - 1}}{40 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b \operatorname{asec}{\left (c x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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