Optimal. Leaf size=57 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {377, 212, 206, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rubi steps
\begin {align*} \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a-(a b-a (-a+b)) x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 a}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.84 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int -\frac {1}{{\left ({\left (a - b\right )} x^{4} - a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-\left (a -b \right ) x^{4}+a \right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ -\int \frac {1}{{\left ({\left (a - b\right )} x^{4} - a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^4+a\right )}^{1/4}\,\left (a-x^4\,\left (a-b\right )\right )} \,d x \]
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 \[ - \int \frac {1}{a x^{4} \sqrt [4]{a + b x^{4}} - a \sqrt [4]{a + b x^{4}} - b x^{4} \sqrt [4]{a + b x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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