Optimal. Leaf size=263 \[ -\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]
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Rubi [A]
time = 1.30, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1307, 222,
1706, 385, 211} \begin {gather*} \frac {\left (-\frac {-2 a c+b^2+b c}{\sqrt {b^2-4 a c}}+b+c\right ) \text {ArcTan}\left (\frac {x \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}{\sqrt {1-x^2} \sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {-2 a c+b^2+b c}{\sqrt {b^2-4 a c}}+b+c\right ) \text {ArcTan}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b+2 c}}{\sqrt {1-x^2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {\text {ArcSin}(x)}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 222
Rule 385
Rule 1307
Rule 1706
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=-\frac {\int \frac {1}{\sqrt {1-x^2}} \, dx}{c}-\frac {\int \frac {-a-(b+c) x^2}{\sqrt {1-x^2} \left (a+b x^2+c x^4\right )} \, dx}{c}\\ &=-\frac {\sin ^{-1}(x)}{c}-\frac {\int \left (\frac {-b-c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}}{\sqrt {1-x^2} \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}+\frac {-b-c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}}{\sqrt {1-x^2} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}\right ) \, dx}{c}\\ &=-\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\sqrt {1-x^2} \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )} \, dx}{c}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\sqrt {1-x^2} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )} \, dx}{c}\\ &=-\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-b-2 c+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{c}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-b-2 c-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{c}\\ &=-\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.43, size = 412, normalized size = 1.57 \begin {gather*} -\frac {8 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-x^2}}\right )+\text {RootSum}\left [a+4 a \text {$\#$1}^2+4 b \text {$\#$1}^2+6 a \text {$\#$1}^4+8 b \text {$\#$1}^4+16 c \text {$\#$1}^4+4 a \text {$\#$1}^6+4 b \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right )-3 a \log (x) \text {$\#$1}^2-4 b \log (x) \text {$\#$1}^2-4 c \log (x) \text {$\#$1}^2+3 a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a \log (x) \text {$\#$1}^4-4 b \log (x) \text {$\#$1}^4-4 c \log (x) \text {$\#$1}^4+3 a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-a \log (x) \text {$\#$1}^6+a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{a \text {$\#$1}+b \text {$\#$1}+3 a \text {$\#$1}^3+4 b \text {$\#$1}^3+8 c \text {$\#$1}^3+3 a \text {$\#$1}^5+3 b \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 c} \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 = 0.13, size = 175, normalized size = 0.67
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )}{c}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}+\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \,\textit {\_R}^{6}+\left (4 c +3 a +4 b \right ) \textit {\_R}^{4}+\left (4 c +3 a +4 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\frac {\sqrt {-x^{2}+1}-1}{x}-\textit {\_R} \right )}{\textit {\_R}^{7} a +3 \textit {\_R}^{5} a +3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +8 c \,\textit {\_R}^{3}+\textit {\_R} a +\textit {\_R} b}}{4 c}\) | \(175\) |
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. 1430 vs.
\(2 (223) = 446\).
time = 0.73, size = 1430, normalized size = 5.44 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c + \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x - {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c - \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x - {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c + \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x + {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c - \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x + {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}{2 \, c} \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 {x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, 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. 3580 vs.
\(2 (223) = 446\).
time = 5.38, size = 3580, normalized size = 13.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.27, size = 870, normalized size = 3.31 \begin {gather*} -\frac {\mathrm {asin}\left (x\right )}{c}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (2\,a\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (8\,a\,c-2\,b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (2\,a\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\left (8\,a\,c-2\,b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (2\,a\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (8\,a\,c-2\,b^2\right )}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (2\,a\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\left (8\,a\,c-2\,b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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