Optimal. Leaf size=83 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b} \]
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Rubi [A] time = 0.05, antiderivative size = 83, 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]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \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 (2 b+a x^4\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{2 b-a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.80 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 83, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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} + 2 \, 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}+2 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} + 2 \, 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 )}^{1/4}\,\left (a\,x^4+2\,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 {1}{\sqrt [4]{a x^{4} + b} \left (a x^{4} + 2 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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