Optimal. Leaf size=35 \[ -\tan ^{-1}\left (\frac {c x}{\sqrt [4]{a x^6+b}}\right )-\tanh ^{-1}\left (\frac {c x}{\sqrt [4]{a x^6+b}}\right ) \]
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Rubi [F] time = 1.14, 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+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx &=\int \left (\frac {c}{\sqrt [4]{b+a x^6}}-\frac {3 b c-c^5 x^4}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )}\right ) \, dx\\ &=c \int \frac {1}{\sqrt [4]{b+a x^6}} \, dx-\int \frac {3 b c-c^5 x^4}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx\\ &=\frac {\left (c \sqrt [4]{1+\frac {a x^6}{b}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {a x^6}{b}}} \, dx}{\sqrt [4]{b+a x^6}}-\int \left (\frac {c^5 x^4}{\left (-b+c^4 x^4-a x^6\right ) \sqrt [4]{b+a x^6}}+\frac {3 b c}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )}\right ) \, dx\\ &=\frac {c x \sqrt [4]{1+\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-\frac {a x^6}{b}\right )}{\sqrt [4]{b+a x^6}}-(3 b c) \int \frac {1}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx-c^5 \int \frac {x^4}{\left (-b+c^4 x^4-a x^6\right ) \sqrt [4]{b+a x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.87, size = 37, normalized size = 1.06 \begin {gather*} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^6}}{c x}\right )-\tanh ^{-1}\left (\frac {c x}{\sqrt [4]{b+a x^6}}\right ) \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 {a c x^{6} - 2 \, b c}{{\left (c^{4} x^{4} - a x^{6} - b\right )} {\left (a x^{6} + 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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a c \,x^{6}-2 b c}{\left (a \,x^{6}+b \right )^{\frac {1}{4}} \left (-c^{4} x^{4}+a \,x^{6}+b \right )}\, dx\]
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 \begin {gather*} -\int \frac {a c x^{6} - 2 \, b c}{{\left (c^{4} x^{4} - a x^{6} - b\right )} {\left (a x^{6} + 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 = -0.03 \begin {gather*} \int -\frac {2\,b\,c-a\,c\,x^6}{{\left (a\,x^6+b\right )}^{1/4}\,\left (-c^4\,x^4+a\,x^6+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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