Optimal. Leaf size=64 \[ \frac{1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{6 c^3} \]
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Rubi [A] time = 0.0265671, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5220, 271, 191} \[ \frac{1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{6 c^3} \]
Antiderivative was successfully verified.
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Rule 5220
Rule 271
Rule 191
Rubi steps
\begin{align*} \int x^3 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \int \frac{x^2}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{4 c}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}+\frac{1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{6 c^3}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x}{6 c^3}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}+\frac{1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.100045, size = 62, normalized size = 0.97 \[ \frac{a x^4}{4}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left (-\frac{x}{6 c^3}-\frac{x^3}{12 c}\right )+\frac{1}{4} b x^4 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.162, size = 74, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{{c}^{4}{x}^{4}a}{4}}+b \left ({\frac{{c}^{4}{x}^{4}{\rm arcsec} \left (cx\right )}{4}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ({c}^{2}{x}^{2}+2 \right ) }{12\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976702, size = 81, normalized size = 1.27 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arcsec}\left (c x\right ) - \frac{c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 3 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.66535, size = 119, normalized size = 1.86 \begin{align*} \frac{3 \, b c^{4} x^{4} \operatorname{arcsec}\left (c x\right ) + 3 \, a c^{4} x^{4} -{\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt{c^{2} x^{2} - 1}}{12 \, c^{4}} \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^{3} \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^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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