Optimal. Leaf size=159 \[ -\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{2 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{2 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}} \]
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Rubi [A] time = 0.16, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {261, 6309, 12, 1252, 848, 50, 63, 208} \[ -\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{2 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{2 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 261
Rule 848
Rule 1252
Rule 6309
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int -\frac {\sqrt {1-c^4 x^4}}{2 c^4 x \sqrt {1-c^2 x^2}} \, dx}{c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {1-c^4 x^4}}{x \sqrt {1-c^2 x^2}} \, dx}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-c^4 x^2}}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+c^2 x}}{x} \, dx,x,x^2\right )}{4 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{2 c^7 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}+\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 140, normalized size = 0.88 \[ -\frac {a \sqrt {1-c^4 x^4}+\frac {b \sqrt {1-c^4 x^4}}{\sqrt {\frac {1-c x}{c x+1}} (c x+1)}+b \log \left (-\sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^4 x^4}-c x+1\right )+b \sqrt {1-c^4 x^4} \text {sech}^{-1}(c x)-b \log (x (1-c x))}{2 c^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 279, normalized size = 1.75 \[ \frac {2 \, \sqrt {-c^{4} x^{4} + 1} b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (b c^{2} x^{2} - b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} x^{2} - b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) + {\left (b c^{2} x^{2} - b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) - 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (a c^{2} x^{2} - a\right )}}{4 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}{\sqrt {-c^{4} x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b {\left (\frac {{\left (c^{4} x^{4} - 1\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{\sqrt {c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {-c x + 1} c^{4}} - 2 \, \int \frac {2 \, c^{2} x^{5} \log \relax (c) + 4 \, c^{2} x^{5} \log \left (\sqrt {x}\right ) + {\left (4 \, c^{2} x^{5} \log \left (\sqrt {x}\right ) + {\left (c^{2} x^{2} {\left (2 \, \log \relax (c) + 1\right )} + 1\right )} x^{3}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{2 \, {\left (c^{2} x^{2} e^{\left (\log \left (c x + 1\right ) + \log \left (-c x + 1\right )\right )} + c^{2} x^{2} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x}\right )} - \frac {\sqrt {-c^{4} x^{4} + 1} a}{2 \, c^{4}} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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