Optimal. Leaf size=77 \[ \frac{1}{4} x^4 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^2 \sqrt{1-c x}}{12 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{b \sqrt{1-c x}}{6 c^4 \sqrt{\frac{1}{c x+1}}} \]
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Rubi [A] time = 0.0307868, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6283, 100, 12, 74} \[ \frac{1}{4} x^4 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^2 \sqrt{1-c x}}{12 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{b \sqrt{1-c x}}{6 c^4 \sqrt{\frac{1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 6283
Rule 100
Rule 12
Rule 74
Rubi steps
\begin{align*} \int x^3 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{4} x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x^2 \sqrt{1-c x}}{12 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{4} x^4 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{2 x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{12 c^2}\\ &=-\frac{b x^2 \sqrt{1-c x}}{12 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{4} x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{6 c^2}\\ &=-\frac{b \sqrt{1-c x}}{6 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^2 \sqrt{1-c x}}{12 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{4} x^4 \left (a+b \text{sech}^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0767012, size = 77, normalized size = 1. \[ \frac{a x^4}{4}+b \sqrt{\frac{1-c x}{c x+1}} \left (-\frac{x^2}{12 c^2}-\frac{x}{6 c^3}-\frac{1}{6 c^4}-\frac{x^3}{12 c}\right )+\frac{1}{4} b x^4 \text{sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.187, size = 72, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{{c}^{4}{x}^{4}a}{4}}+b \left ({\frac{{c}^{4}{x}^{4}{\rm arcsech} \left (cx\right )}{4}}-{\frac{cx \left ({c}^{2}{x}^{2}+2 \right ) }{12}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00829, size = 77, normalized size = 1. \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arsech}\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 = 1.98669, size = 193, normalized size = 2.51 \begin{align*} \frac{3 \, b c^{3} x^{4} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 3 \, a c^{3} x^{4} -{\left (b c^{2} x^{3} + 2 \, b x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.8029, size = 68, normalized size = 0.88 \begin{align*} \begin{cases} \frac{a x^{4}}{4} + \frac{b x^{4} \operatorname{asech}{\left (c x \right )}}{4} - \frac{b x^{2} \sqrt{- c^{2} x^{2} + 1}}{12 c^{2}} - \frac{b \sqrt{- c^{2} x^{2} + 1}}{6 c^{4}} & \text{for}\: c \neq 0 \\\frac{x^{4} \left (a + \infty b\right )}{4} & \text{otherwise} \end{cases} \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{arsech}\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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