Optimal. Leaf size=329 \[ -\frac {\left (\sqrt {2 \sqrt {a} \sqrt {b}+c^2}+c\right ) \sqrt {c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}-\sqrt {a} \sqrt {b}-c^2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}-\sqrt {a} \sqrt {b}-c^2}}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c}-\frac {\left (\sqrt {2 \sqrt {a} \sqrt {b}+c^2}-c\right ) \sqrt {c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}+\sqrt {a} \sqrt {b}+c^2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}+\sqrt {a} \sqrt {b}+c^2}}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c} \]
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Rubi [A] time = 0.15, 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 = {2112, 208} \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 208
Rule 2112
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 \operatorname {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 [C] time = 0.95, size = 199, normalized size = 0.60 \begin {gather*} -\frac {i \sqrt {\frac {a x^4}{b}+1} \left (-\Pi \left (\frac {2 i \sqrt {a} \sqrt {b}}{c^2-\sqrt {c^4-4 a b}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )-\Pi \left (\frac {2 i \sqrt {a} \sqrt {b}}{c^2+\sqrt {c^4-4 a b}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )+F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} \sqrt {a x^4+b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.26, 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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fricas [A] time = 0.59, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} - b}{{\left (a x^{4} - c^{2} x^{2} + b\right )} \sqrt {a x^{4} + b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 23, normalized size = 0.07
method | result | size |
elliptic | \(-\frac {\arctanh \left (\frac {\sqrt {a \,x^{4}+b}}{x c}\right )}{c}\) | \(23\) |
default | \(\frac {\sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, \sqrt {a \,x^{4}+b}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-c^{2} \textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-2 b \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a +a \,x^{2}+c^{2}\right )}{\sqrt {c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}+b}}\right )}{\sqrt {c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right ) \sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right )}{\sqrt {b}\, \sqrt {a}}, \frac {\sqrt {-\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b \sqrt {a \,x^{4}+b}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right )}\right )}{4}\) | \(297\) |
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*} \int \frac {a x^{4} - b}{{\left (a x^{4} - c^{2} x^{2} + b\right )} \sqrt {a x^{4} + b}}\,{d x} \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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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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