Optimal. Leaf size=329 \[ -\frac {\left (c+\sqrt {2 \sqrt {a} \sqrt {b}+c^2}\right ) \sqrt {-\sqrt {a} \sqrt {b}-c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-\sqrt {a} \sqrt {b}-c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c}-\frac {\left (-c+\sqrt {2 \sqrt {a} \sqrt {b}+c^2}\right ) \sqrt {\sqrt {a} \sqrt {b}+c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a} \sqrt {b}+c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c} \]
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Rubi [A]
time = 0.12, antiderivative size = 20, normalized size of antiderivative = 0.06, number of steps
used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2137, 214}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {c x}{\sqrt {a x^4+b}}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2137
Rubi steps
\begin {align*} \int \frac {-b+a x^4}{\sqrt {b+a x^4} \left (b-c^2 x^2+a x^4\right )} \, dx &=-\left (b \text {Subst}\left (\int \frac {1}{b-b c^2 x^2} \, dx,x,\frac {x}{\sqrt {b+a x^4}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {c x}{\sqrt {b+a x^4}}\right )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 52, normalized size = 0.16 \begin {gather*} -\frac {\log \left (c x+\sqrt {b+a x^4}\right )}{2 c}+\frac {\log \left (c^2 x-c \sqrt {b+a x^4}\right )}{2 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 = 0.14, size = 297, normalized size = 0.90
method | result | size |
elliptic | \(-\frac {\arctanh \left (\frac {\sqrt {a \,x^{4}+b}}{x c}\right )}{c}\) | \(23\) |
default | \(\frac {\sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, \sqrt {a \,x^{4}+b}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-c^{2} \textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-2 b \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a +a \,x^{2}+c^{2}\right )}{\sqrt {c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}+b}}\right )}{\sqrt {c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right ) \sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right )}{\sqrt {a}\, \sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b \sqrt {a \,x^{4}+b}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right )}\right )}{4}\) | \(297\) |
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 [A]
time = 0.45, size = 51, normalized size = 0.16 \begin {gather*} \frac {\log \left (\frac {a x^{4} + c^{2} x^{2} - 2 \, \sqrt {a x^{4} + b} c x + b}{a x^{4} - c^{2} x^{2} + b}\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 {a x^{4} - b}{\sqrt {a x^{4} + b} \left (a x^{4} + b - c^{2} x^{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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {b-a\,x^4}{\sqrt {a\,x^4+b}\,\left (-c^2\,x^2+a\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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