Optimal. Leaf size=77 \[ \frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b \sqrt {1-c x}}{6 c^4 \sqrt {\frac {1}{c x+1}}}-\frac {b x^2 \sqrt {1-c x}}{12 c^2 \sqrt {\frac {1}{c x+1}}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 12
Rule 74
Rule 100
Rule 6283
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*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 1.00 \[ \frac {a x^4}{4}+b \sqrt {\frac {1-c x}{c x+1}} \left (-\frac {1}{6 c^4}-\frac {x}{6 c^3}-\frac {x^2}{12 c^2}-\frac {x^3}{12 c}\right )+\frac {1}{4} b x^4 \text {sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 90, normalized size = 1.17 \[ \frac {3 \, b c^{3} x^{4} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 3 \, a c^{3} x^{4} - {\left (b c^{2} x^{3} + 2 \, b x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 72, normalized size = 0.94 \[ \frac {\frac {c^{4} x^{4} a}{4}+b \left (\frac {c^{4} x^{4} \mathrm {arcsech}\left (c x \right )}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c^{2} x^{2}+2\right )}{12}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 57, normalized size = 0.74 \[ \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.97, size = 68, normalized size = 0.88 \[ \begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {asech}{\left (c x \right )}}{4} - \frac {b x^{2} \sqrt {- c^{2} x^{2} + 1}}{12 c^{2}} - \frac {b \sqrt {- c^{2} x^{2} + 1}}{6 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4} \left (a + \infty b\right )}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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