Optimal. Leaf size=106 \[ -\frac {\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\& \right ]}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(106)=212\).
time = 0.22, antiderivative size = 455, normalized size of antiderivative = 4.29, number of steps
used = 9, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1442, 385, 218,
214, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 385
Rule 1442
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx &=\frac {2 \int \frac {1}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx}{\sqrt {a^2+4 b}}-\frac {2 \int \frac {1}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx}{\sqrt {a^2+4 b}}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+4 b}-\left (2 b+a \left (-a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b}}-\frac {2 \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+4 b}-\left (2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b}}\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 106, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{4}-b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
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.
[In]
[Out]
________________________________________________________________________________________
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 + x^{8}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 (-x^8+a\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________