Optimal. Leaf size=116 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}-\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]
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Rubi [A]
time = 0.02, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {414}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}-\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \end {gather*}
Antiderivative was successfully verified.
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Rule 414
Rubi steps
\begin {align*} \int \frac {\sqrt {a-b x^4}}{a c+b c x^4} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x \left (\sqrt {a}-\sqrt {b} x^2\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.31, size = 88, normalized size = 0.76 \begin {gather*} \frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (\tan ^{-1}\left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a-b x^4}}\right )-i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-b x^4}}{\sqrt [4]{a} \sqrt [4]{b} x}\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 165, normalized size = 1.42
method | result | size |
default | \(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {\frac {-b \,x^{4}+a}{2 x^{2}}-\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}{\frac {-b \,x^{4}+a}{2 x^{2}}+\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}\right )}{8 \left (a b \right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{\left (a b \right )^{\frac {1}{4}} x}+1\right )}{4 \left (a b \right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{\left (a b \right )^{\frac {1}{4}} x}-1\right )}{4 \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2 c}\) | \(165\) |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {\frac {-b \,x^{4}+a}{2 x^{2}}-\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}{\frac {-b \,x^{4}+a}{2 x^{2}}+\frac {\left (a b \right )^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}}{x}+\sqrt {a b}}\right )}{8 c \left (a b \right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{\left (a b \right )^{\frac {1}{4}} x}+1\right )}{4 c \left (a b \right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{\left (a b \right )^{\frac {1}{4}} x}-1\right )}{4 c \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(171\) |
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. 339 vs.
\(2 (86) = 172\).
time = 4.66, size = 339, normalized size = 2.92 \begin {gather*} -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} \sqrt {-\frac {1}{b}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (b c x^{2} \sqrt {-\frac {1}{b}} + \sqrt {-b x^{4} + a} c\right )} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (-\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} - \sqrt {-b x^{4} + a} a c^{2} \sqrt {-\frac {1}{a b c^{4}}} - 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (-\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {-b x^{4} + a} x^{2}}{b x^{4} + a}\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*} \frac {\int \frac {\sqrt {a - b x^{4}}}{a + b x^{4}}\, dx}{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*} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a-b\,x^4}}{b\,c\,x^4+a\,c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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