Optimal. Leaf size=65 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b+c x^2}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b+c x^2}}\right )}{\sqrt [4]{a}} \]
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Rubi [F] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx &=\int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.43, size = 65, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b+c x^2}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b+c x^2}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 135, normalized size = 2.08 \begin {gather*} \frac {2 \, \arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + c x^{2} - b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {{\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \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 {c x^{2} - 2 \, b}{{\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}} {\left (c x^{2} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c \,x^{2}-2 b}{\left (c \,x^{2}-b \right ) \left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}\, dx \end {gather*}
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 {c x^{2} - 2 \, b}{{\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}} {\left (c x^{2} - b\right )}}\,{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 = -0.02 \begin {gather*} \int \frac {2\,b-c\,x^2}{\left (b-c\,x^2\right )\,{\left (a\,x^4+c\,x^2-b\right )}^{1/4}} \,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 {- 2 b + c x^{2}}{\left (- b + c x^{2}\right ) \sqrt [4]{a x^{4} - b + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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