Optimal. Leaf size=123 \[ -\frac {4 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.29, antiderivative size = 747, normalized size of antiderivative = 6.07, number of steps
used = 19, number of rules used = 11, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2081, 6847,
6857, 226, 2098, 1225, 1713, 214, 6860, 1231, 1721} \begin {gather*} -\frac {4 \sqrt {x} \sqrt {a^2 x^2+b^2} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {2 \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {4 \sqrt {a} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \left (3 \sqrt {-a^2}+\sqrt {3} a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {4 \sqrt {-a^2} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (\sqrt {3} \sqrt {-a^2}+3 a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \Pi \left (\frac {1}{4};2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \Pi \left (\frac {1}{4};2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (\sqrt {3} \sqrt {-a^2}+3 a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {2} \sqrt {x} \sqrt {a^2 x^2+b^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 226
Rule 1225
Rule 1231
Rule 1713
Rule 1721
Rule 2081
Rule 2098
Rule 6847
Rule 6857
Rule 6860
Rubi steps
\begin {align*} \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {b^3+a^3 x^3}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (-b^3+a^3 x^3\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b^3+a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (-b^3+a^3 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}+\frac {2 b^3}{\sqrt {b^2+a^2 x^4} \left (-b^3+a^3 x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (-b^3+a^3 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{3 b^2 \left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {-2 b-a x^2}{3 b^2 \sqrt {b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-2 b-a x^2}{\sqrt {b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b+a x^2}{\left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{b-2 a b^2 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \left (-a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 a \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 a \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {4 \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {a} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \left (\sqrt {3} a+3 \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^2} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 129, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \left (4 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{3 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 387, normalized size = 3.15
method | result | size |
default | \(\frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a -2 b \right ) \left (i \underline {\hspace {1.25 ex}}\alpha a +i b +b \right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha a -i b +b}{b}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha a +b \right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{3 a}+\frac {2 i b^{2} \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{3 a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}\) | \(387\) |
elliptic | \(\text {Expression too large to display}\) | \(875\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 203 vs.
\(2 (95) = 190\).
time = 0.42, size = 423, normalized size = 3.44 \begin {gather*} \left [\frac {\sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} + 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 12 \, a b^{3} x + b^{4} - 4 \, \sqrt {2} {\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {\frac {1}{a b}}}{a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right )}{12 \, a b}, \frac {\sqrt {2} a b \sqrt {-\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {-\frac {1}{a b}}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right ) - 2 \, \sqrt {-a b} \log \left (\frac {a^{4} x^{4} - 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {-a b}}{a^{4} x^{4} + 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}}\right )}{6 \, a b}\right ] \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 \frac {\left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \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(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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