Optimal. Leaf size=65 \[ -\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{c^2} \]
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Rubi [A]
time = 0.06, 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 = {6420, 5559,
4269, 3556} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4269
Rule 5559
Rule 6420
Rubi steps
\begin {align*} \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=-\frac {\text {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 \text {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 \text {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 = 0.14, size = 112, normalized size = 1.72 \begin {gather*} \frac {a \left (a c^2 x^2-2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )-2 b \left (-a c^2 x^2+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 c^2 x^2 \text {sech}^{-1}(c x)^2-2 b^2 \log (c x)}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(172\) vs.
\(2(61)=122\).
time = 0.44, size = 173, normalized size = 2.66
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsech}\left (c x \right )+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+2 a b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{2} x^{2}}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{2}\right )}{c^{2}}\) | \(173\) |
default | \(\frac {\frac {a^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsech}\left (c x \right )+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+2 a b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{2} x^{2}}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{2}\right )}{c^{2}}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 84, normalized size = 1.29 \begin {gather*} \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 \left (x\right )}{c^{2}}\right )} b^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs.
\(2 (61) = 122\).
time = 0.42, size = 205, normalized size = 3.15 \begin {gather*} \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 \left (x\right ) + 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 99, normalized size = 1.52 \begin {gather*} \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 {\left (x \right )}}{c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \infty b\right )^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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