Optimal. Leaf size=450 \[ -\frac{501 a^8 x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{128 (1-a x)^4 (a x+1)^4}+\frac{373 a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{192 (1-a x)^4 (a x+1)^3}+\frac{501 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{640 (1-a x)^4 (a x+1)^2}+\frac{661 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{1680 (1-a x)^4 (a x+1)}+\frac{295 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{1344 (1-a x)^4}-\frac{127 a^3 x^4 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{420 (1-a x)^4}+\frac{71 a^2 x^3 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{336 (1-a x)^3}-\frac{a x^2 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{28 (1-a x)^2}-\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{8 (1-a x)}+\frac{2 a^8 x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \sin ^{-1}(a x)}{(1-a x)^{9/2} (a x+1)^{9/2}}+\frac{245 a^8 x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{128 (1-a x)^{9/2} (a x+1)^{9/2}} \]
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Rubi [A] time = 0.55677, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac{501 a^8 x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{128 (1-a x)^4 (a x+1)^4}+\frac{373 a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{192 (1-a x)^4 (a x+1)^3}+\frac{501 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{640 (1-a x)^4 (a x+1)^2}+\frac{661 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{1680 (1-a x)^4 (a x+1)}+\frac{295 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{1344 (1-a x)^4}-\frac{127 a^3 x^4 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{420 (1-a x)^4}+\frac{71 a^2 x^3 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{336 (1-a x)^3}-\frac{a x^2 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{28 (1-a x)^2}-\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{8 (1-a x)}+\frac{2 a^8 x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \sin ^{-1}(a x)}{(1-a x)^{9/2} (a x+1)^{9/2}}+\frac{245 a^8 x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{128 (1-a x)^{9/2} (a x+1)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6159
Rule 6129
Rule 97
Rule 149
Rule 154
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \, dx &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^{9/2} (1+a x)^{9/2}}{x^9} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1-a x)^{7/2} (1+a x)^{11/2}}{x^9} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1-a x)^{5/2} (1+a x)^{9/2} \left (2 a-9 a^2 x\right )}{x^8} \, dx}{8 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1-a x)^{3/2} (1+a x)^{9/2} \left (-71 a^2+61 a^3 x\right )}{x^7} \, dx}{56 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{\sqrt{1-a x} (1+a x)^{9/2} \left (508 a^3-295 a^4 x\right )}{x^6} \, dx}{336 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1+a x)^{9/2} \left (-1475 a^4+967 a^5 x\right )}{x^5 \sqrt{1-a x}} \, dx}{1680 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1+a x)^{7/2} \left (-7932 a^5+5343 a^6 x\right )}{x^4 \sqrt{1-a x}} \, dx}{6720 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1+a x)^{5/2} \left (-31563 a^6+23961 a^7 x\right )}{x^3 \sqrt{1-a x}} \, dx}{20160 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac{501 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1+a x)^{3/2} \left (-78330 a^7+79485 a^8 x\right )}{x^2 \sqrt{1-a x}} \, dx}{40320 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac{373 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac{501 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{\sqrt{1+a x} \left (-77175 a^8+157815 a^9 x\right )}{x \sqrt{1-a x}} \, dx}{40320 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac{501 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac{373 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac{501 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{77175 a^9-80640 a^{10} x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{40320 a (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac{501 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac{373 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac{501 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}-\frac{\left (245 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{128 (1-a x)^{9/2} (1+a x)^{9/2}}+\frac{\left (2 a^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac{501 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac{373 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac{501 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{\left (245 a^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}}+\frac{\left (2 a^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac{295 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac{501 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac{373 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac{501 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac{661 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac{127 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac{71 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac{2 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9 \sin ^{-1}(a x)}{(1-a x)^{9/2} (1+a x)^{9/2}}+\frac{245 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9 \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.185555, size = 166, normalized size = 0.37 \[ -\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (13440 a^8 x^8-45056 a^7 x^7+14595 a^6 x^6+31232 a^5 x^5+770 a^4 x^4-16896 a^3 x^3-4760 a^2 x^2+3840 a x+1680\right )+26880 a^8 x^8 \log \left (\sqrt{a^2 x^2-1}+a x\right )+25725 a^8 x^8 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{13440 a^8 x^7 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.274, size = 965, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a x + 1\right )}^{2}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43752, size = 1129, normalized size = 2.51 \begin{align*} \left [\frac{53760 \, a^{7} \sqrt{-c} c^{4} x^{7} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 25725 \, a^{7} \sqrt{-c} c^{4} x^{7} \log \left (-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \,{\left (13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{26880 \, a^{8} x^{7}}, -\frac{25725 \, a^{7} c^{\frac{9}{2}} x^{7} \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 13440 \, a^{7} c^{\frac{9}{2}} x^{7} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) +{\left (13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{13440 \, a^{8} x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 57.1708, size = 954, normalized size = 2.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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