Optimal. Leaf size=124 \[ -\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{6 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{3 c^4}-\frac {b^2 x^2}{12 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6285, 5451, 4185, 4184, 3475} \[ -\frac {b x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{6 c^2}-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^4}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 x^2}{12 c^2}-\frac {b^2 \log (x)}{3 c^4} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 4185
Rule 5451
Rule 6285
Rubi steps
\begin {align*} \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \text {sech}^4(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^4}\\ &=\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b \operatorname {Subst}\left (\int (a+b x) \text {sech}^4(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 c^4}\\ &=-\frac {b^2 x^2}{12 c^2}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{6 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b \operatorname {Subst}\left (\int (a+b x) \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{3 c^4}\\ &=-\frac {b^2 x^2}{12 c^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{6 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {b^2 \operatorname {Subst}\left (\int \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{3 c^4}\\ &=-\frac {b^2 x^2}{12 c^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{6 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{3 c^4}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 212, normalized size = 1.71 \[ -\frac {-3 a^2 c^4 x^4+2 a b c^3 x^3 \sqrt {\frac {1-c x}{c x+1}}+2 a b c^2 x^2 \sqrt {\frac {1-c x}{c x+1}}+2 b \text {sech}^{-1}(c x) \left (b \sqrt {\frac {1-c x}{c x+1}} \left (c^3 x^3+c^2 x^2+2 c x+2\right )-3 a c^4 x^4\right )+4 a b c x \sqrt {\frac {1-c x}{c x+1}}+4 a b \sqrt {\frac {1-c x}{c x+1}}-3 b^2 c^4 x^4 \text {sech}^{-1}(c x)^2+b^2 c^2 x^2+4 b^2 \log (x)}{12 c^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 244, normalized size = 1.97 \[ \frac {3 \, b^{2} c^{4} x^{4} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, a^{2} c^{4} x^{4} - 6 \, a b c^{4} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - b^{2} c^{2} x^{2} - 4 \, b^{2} \log \relax (x) + 2 \, {\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{4} - {\left (b^{2} c^{3} x^{3} + 2 \, b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (a b c^{3} x^{3} + 2 \, a b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, c^{4}} \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.72, size = 264, normalized size = 2.13 \[ \frac {a^{2} x^{4}}{4}-\frac {b^{2} \mathrm {arcsech}\left (c x \right )}{3 c^{4}}+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{2} x^{4}}{4}-\frac {b^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \mathrm {arcsech}\left (c x \right ) x^{3}}{6 c}-\frac {b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{3 c^{3}}-\frac {b^{2} x^{2}}{12 c^{2}}+\frac {b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{3 c^{4}}+\frac {a b \,x^{4} \mathrm {arcsech}\left (c x \right )}{2}-\frac {a b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{3}}{6 c}-\frac {a b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}}{3 c^{3}} \]
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 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} a b + b^{2} \int x^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}\,{d x} \]
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 \[ \int x^3\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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