Optimal. Leaf size=65 \[ -\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{c^2} \]
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Rubi [A] time = 0.08, antiderivative size = 65, 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 = {6285, 5451, 4184, 3475} \[ -\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 5451
Rule 6285
Rubi steps
\begin {align*} \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \text {sech}^2(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^2}\\ &=\frac {1}{2} x^2 \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 )}{c^2}\\ &=-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}+\frac {1}{2} x^2 \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 )}{c^2}\\ &=-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{c^2}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 112, normalized size = 1.72 \[ \frac {a \left (a c^2 x^2-2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1)\right )-2 b \text {sech}^{-1}(c x) \left (b \sqrt {\frac {1-c x}{c x+1}} (c x+1)-a c^2 x^2\right )+b^2 c^2 x^2 \text {sech}^{-1}(c x)^2-2 b^2 \log (c x)}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 205, normalized size = 3.15 \[ \frac {b^{2} c^{2} x^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + a^{2} c^{2} x^{2} - 2 \, a b c^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, b^{2} \log \relax (x) + 2 \, {\left (a b c^{2} x^{2} - b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b c^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c^{2}} \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 168, normalized size = 2.58 \[ \frac {a^{2} x^{2}}{2}-\frac {b^{2} \mathrm {arcsech}\left (c x \right )}{c^{2}}+\frac {x^{2} b^{2} \mathrm {arcsech}\left (c x \right )^{2}}{2}-\frac {b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{c}+\frac {b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}+a b \,\mathrm {arcsech}\left (c x \right ) x^{2}-\frac {a b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 84, normalized size = 1.29 \[ \frac {1}{2} \, b^{2} x^{2} \operatorname {arsech}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} a b - {\left (\frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right )}{c} + \frac {\log \relax (x)}{c^{2}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.17, size = 99, normalized size = 1.52 \[ \begin {cases} \frac {a^{2} x^{2}}{2} + a b x^{2} \operatorname {asech}{\left (c x \right )} - \frac {a b \sqrt {- c^{2} x^{2} + 1}}{c^{2}} + \frac {b^{2} x^{2} \operatorname {asech}^{2}{\left (c x \right )}}{2} - \frac {b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asech}{\left (c x \right )}}{c^{2}} - \frac {b^{2} \log {\relax (x )}}{c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \infty b\right )^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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