Optimal. Leaf size=87 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
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Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {377, 212, 206, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{b-2 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 0.80 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 87, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 172.99, size = 431, normalized size = 4.95 \begin {gather*} -\frac {1}{2} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x^{4} - b\right )}^{\frac {3}{4}} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + {\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {a x^{4} - b} a b x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (3 \, a^{2} b^{3} x^{4} - a b^{4}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{a b^{4}}}}\right )}}{a x^{4} + b}\right ) + \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) \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 {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{4}+b \right )}\, 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 {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{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 = -0.01 \begin {gather*} \int \frac {1}{\left (a\,x^4+b\right )\,{\left (a\,x^4-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 {1}{\sqrt [4]{a x^{4} - b} \left (a x^{4} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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