Optimal. Leaf size=385 \[ -\frac{663 d^{19/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.445966, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {28, 288, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{663 d^{19/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{19/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (17 b^4 d^2\right ) \int \frac{(d x)^{15/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{320} \left (221 b^2 d^4\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{\left (663 d^6\right ) \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}+\frac{\left (663 d^8\right ) \int \frac{(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (663 d^{10}\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^4}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (663 d^9\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 b^4}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (663 d^8\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d-\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{8192 \sqrt{a} b^4}+\frac{\left (663 d^8\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d+\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{8192 \sqrt{a} b^4}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (663 d^{19/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}-\frac{\left (663 d^{19/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{\left (663 d^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{16384 \sqrt{a} b^{11/2}}+\frac{\left (663 d^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{16384 \sqrt{a} b^{11/2}}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}-\frac{663 d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{\left (663 d^{19/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}-\frac{\left (663 d^{19/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}\\ &=-\frac{d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left (a+b x^2\right )}-\frac{663 d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{3/4} b^{21/4}}\\ \end{align*}
Mathematica [A] time = 0.204056, size = 381, normalized size = 0.99 \[ \frac{d^9 \sqrt{d x} \left (-\frac{72417280 a^2 b^{9/4} x^4}{\left (a+b x^2\right )^5}-\frac{43450368 a^3 b^{5/4} x^2}{\left (a+b x^2\right )^5}+\frac{848640 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac{678912 a^3 \sqrt [4]{b}}{\left (a+b x^2\right )^4}-\frac{10862592 a^4 \sqrt [4]{b}}{\left (a+b x^2\right )^5}-\frac{765765 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} \sqrt{x}}+\frac{765765 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} \sqrt{x}}-\frac{1531530 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4} \sqrt{x}}+\frac{1531530 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} \sqrt{x}}-\frac{25231360 b^{17/4} x^8}{\left (a+b x^2\right )^5}-\frac{61276160 a b^{13/4} x^6}{\left (a+b x^2\right )^5}+\frac{2042040 \sqrt [4]{b}}{a+b x^2}+\frac{1166880 a \sqrt [4]{b}}{\left (a+b x^2\right )^2}\right )}{37847040 b^{21/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 344, normalized size = 0.9 \begin{align*} -{\frac{663\,{d}^{19}{a}^{4}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{5}}\sqrt{dx}}-{\frac{1989\,{d}^{17}{a}^{3}}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{4}} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{9061\,{d}^{15}{a}^{2}}{6144\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}-{\frac{527\,{d}^{13}a}{384\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{13}{2}}}}-{\frac{7529\,{d}^{11}}{12288\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{17}{2}}}}+{\frac{663\,{d}^{9}\sqrt{2}}{32768\,{b}^{5}a}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({ \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{663\,{d}^{9}\sqrt{2}}{16384\,{b}^{5}a}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{663\,{d}^{9}\sqrt{2}}{16384\,{b}^{5}a}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71322, size = 1127, normalized size = 2.93 \begin{align*} \frac{39780 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{3}{4}} \sqrt{d x} a^{2} b^{16} d^{9} - \sqrt{d^{19} x + \sqrt{-\frac{d^{38}}{a^{3} b^{21}}} a^{2} b^{10}} \left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{3}{4}} a^{2} b^{16}}{d^{38}}\right ) + 9945 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{1}{4}} \log \left (663 \, \sqrt{d x} d^{9} + 663 \, \left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{1}{4}} a b^{5}\right ) - 9945 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{1}{4}} \log \left (663 \, \sqrt{d x} d^{9} - 663 \, \left (-\frac{d^{38}}{a^{3} b^{21}}\right )^{\frac{1}{4}} a b^{5}\right ) - 4 \,{\left (37645 \, b^{4} d^{9} x^{8} + 84320 \, a b^{3} d^{9} x^{6} + 90610 \, a^{2} b^{2} d^{9} x^{4} + 47736 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt{d x}}{245760 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17475, size = 463, normalized size = 1.2 \begin{align*} \frac{1}{491520} \, d^{8}{\left (\frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{6}} + \frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{6}} + \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \log \left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{6}} - \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \log \left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{6}} - \frac{8 \,{\left (37645 \, \sqrt{d x} b^{4} d^{11} x^{8} + 84320 \, \sqrt{d x} a b^{3} d^{11} x^{6} + 90610 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{4} + 47736 \, \sqrt{d x} a^{3} b d^{11} x^{2} + 9945 \, \sqrt{d x} a^{4} d^{11}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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