Optimal. Leaf size=459 \[ -\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 331,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (a b+b^2 x^2\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{a d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{a^2 d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (2 b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{a^2 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b^{3/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{a^2 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (b^{3/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{a^2 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^2\right ) \text {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 )}{2 a^2 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^2\right ) \text {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 )}{2 a^2 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \text {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 )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 \left (a+b x^2\right )}{5 a d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 b \left (a+b x^2\right )}{a^2 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} a^{9/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 160, normalized size = 0.35 \begin {gather*} -\frac {x \left (a+b x^2\right ) \left (4 \sqrt [4]{a} \left (a-5 b x^2\right )+5 \sqrt {2} b^{5/4} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} b^{5/4} x^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{10 a^{9/4} (d x)^{7/2} \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 251, normalized size = 0.55
method | result | size |
risch | \(-\frac {2 \left (-5 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 a^{2} \sqrt {d x}\, x^{2} d^{3} \left (b \,x^{2}+a \right )}+\frac {\left (\frac {b \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{4 a^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{d^{3} \left (b \,x^{2}+a \right )}\) | \(243\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-5 b \sqrt {2}\, \left (d x \right )^{\frac {5}{2}} \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )-10 b \sqrt {2}\, \left (d x \right )^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-10 b \sqrt {2}\, \left (d x \right )^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-40 b \,d^{2} x^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+8 a \,d^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}\right )}{20 d^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a^{2} \left (d x \right )^{\frac {5}{2}} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\) | \(251\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 259, normalized size = 0.56 \begin {gather*} \frac {\frac {5 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{2} d^{2}} + \frac {8 \, {\left (5 \, b d^{2} x^{2} - a d^{2}\right )}}{\left (d x\right )^{\frac {5}{2}} a^{2} d^{2}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 253, normalized size = 0.55 \begin {gather*} -\frac {20 \, a^{2} d^{4} x^{3} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{2} b^{4} d^{3} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-a^{5} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{9} d^{14}}} + b^{8} d x} a^{2} d^{3} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {1}{4}}}{b^{5}}\right ) - 5 \, a^{2} d^{4} x^{3} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {1}{4}} \log \left (a^{7} d^{11} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{4}\right ) + 5 \, a^{2} d^{4} x^{3} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {1}{4}} \log \left (-a^{7} d^{11} \left (-\frac {b^{5}}{a^{9} d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{4}\right ) - 4 \, {\left (5 \, b x^{2} - a\right )} \sqrt {d x}}{10 \, a^{2} d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.05, size = 284, normalized size = 0.62 \begin {gather*} \frac {1}{20} \, {\left (\frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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^{3} b d^{5}} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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^{3} b d^{5}} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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^{3} b d^{5}} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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^{3} b d^{5}} + \frac {8 \, {\left (5 \, b d^{2} x^{2} - a d^{2}\right )}}{\sqrt {d x} a^{2} d^{5} x^{2}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{7/2}\,\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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