Optimal. Leaf size=56 \[ -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 56, 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 = {5330, 4494,
4269, 3556} \begin {gather*} -\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-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 4494
Rule 5330
Rubi steps
\begin {align*} \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int (a+b x)^2 \sec ^2(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {b \text {Subst}\left (\int (a+b x) \sec ^2(x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 \text {Subst}\left (\int \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 90, normalized size = 1.61 \begin {gather*} \frac {a c x \left (-2 b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right )+2 b c x \left (-b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right ) \sec ^{-1}(c x)+b^2 c^2 x^2 \sec ^{-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. \(127\) vs.
\(2(52)=104\).
time = 0.26, size = 128, normalized size = 2.29
method | result | size |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+\frac {b^{2} \mathrm {arcsec}\left (c x \right )^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-b^{2} \ln \left (\frac {1}{c x}\right )+2 a b \left (\frac {c^{2} x^{2} \mathrm {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(128\) |
default | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+\frac {b^{2} \mathrm {arcsec}\left (c x \right )^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-b^{2} \ln \left (\frac {1}{c x}\right )+2 a b \left (\frac {c^{2} x^{2} \mathrm {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 87, normalized size = 1.55 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arcsec}\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 {arcsec}\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. 111 vs.
\(2 (52) = 104\).
time = 2.15, size = 111, normalized size = 1.98 \begin {gather*} \frac {b^{2} c^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + a^{2} c^{2} x^{2} + 4 \, a b c^{2} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, b^{2} \log \left (x\right ) + 2 \, {\left (a b c^{2} x^{2} - a b c^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arcsec}\left (c x\right ) + a b\right )}}{2 \, c^{2}} \end {gather*}
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 \begin {gather*} \int x \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2181 vs.
\(2 (52) = 104\).
time = 0.56, size = 2181, normalized size = 38.95 \begin {gather*} \text {Too large to display} \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 {acos}\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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