Optimal. Leaf size=143 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8 b^4-2 \text {$\#$1}^4 a^4 b^4-\text {$\#$1}^4 c+a^4 c\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a^4 x^4-b^4}-\text {$\#$1} x\right )-\text {$\#$1}^4 \log (x)-a^4 \log \left (\sqrt [4]{a^4 x^4-b^4}-\text {$\#$1} x\right )+a^4 \log (x)}{-2 \text {$\#$1}^5 b^4+2 \text {$\#$1} a^4 b^4+\text {$\#$1} c}\& \right ]}{4 b^4} \]
________________________________________________________________________________________
Rubi [B] time = 0.46, antiderivative size = 601, normalized size of antiderivative = 4.20, number of steps used = 9, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1428, 377, 212, 208, 205} \begin {gather*} -\frac {a^7 \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}-\frac {a^7 \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}-\frac {a^7 \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}-\frac {a^7 \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 208
Rule 212
Rule 377
Rule 1428
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx &=\frac {\left (2 a^8\right ) \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-c-\sqrt {4 a^8 b^8+c^2}+2 a^8 x^4\right )} \, dx}{\sqrt {4 a^8 b^8+c^2}}-\frac {\left (2 a^8\right ) \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-c+\sqrt {4 a^8 b^8+c^2}+2 a^8 x^4\right )} \, dx}{\sqrt {4 a^8 b^8+c^2}}\\ &=\frac {\left (2 a^8\right ) \operatorname {Subst}\left (\int \frac {1}{-c-\sqrt {4 a^8 b^8+c^2}-\left (2 a^8 b^4+a^4 \left (-c-\sqrt {4 a^8 b^8+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2}}-\frac {\left (2 a^8\right ) \operatorname {Subst}\left (\int \frac {1}{-c+\sqrt {4 a^8 b^8+c^2}-\left (2 a^8 b^4+a^4 \left (-c+\sqrt {4 a^8 b^8+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2}}\\ &=-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {4 a^8 b^8+c^2}}-a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {4 a^8 b^8+c^2}}+a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {4 a^8 b^8+c^2}}-a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {4 a^8 b^8+c^2}}+a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {c+\sqrt {4 a^8 b^8+c^2}}}\\ &=-\frac {a^7 \tan ^{-1}\left (\frac {a \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{-c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (-c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^7 \tan ^{-1}\left (\frac {a \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^7 \tanh ^{-1}\left (\frac {a \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{-c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (-c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^7 \tanh ^{-1}\left (\frac {a \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.13, size = 542, normalized size = 3.79 \begin {gather*} -\frac {a^7 \left (\frac {\tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}+\frac {\tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}+\frac {\tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}+\frac {\tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}\right )}{\sqrt {4 a^8 b^8+c^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 144, normalized size = 1.01 \begin {gather*} -\frac {\text {RootSum}\left [a^4 c-2 a^4 b^4 \text {$\#$1}^4-c \text {$\#$1}^4+b^4 \text {$\#$1}^8\&,\frac {-a^4 \log (x)+a^4 \log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a^4 b^4 \text {$\#$1}-c \text {$\#$1}+2 b^4 \text {$\#$1}^5}\&\right ]}{4 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{8} x^{8} - b^{8} - c x^{4}\right )} {\left (a^{4} x^{4} - b^{4}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{4} x^{4}-b^{4}\right )^{\frac {1}{4}} \left (a^{8} x^{8}-b^{8}-c \,x^{4}\right )}\, dx\]
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*} \int \frac {1}{{\left (a^{8} x^{8} - b^{8} - c x^{4}\right )} {\left (a^{4} x^{4} - b^{4}\right )}^{\frac {1}{4}}}\,{d x} \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^4\,x^4-b^4\right )}^{1/4}\,\left (-a^8\,x^8+b^8+c\,x^4\right )} \,d x \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]{\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a^{8} x^{8} - b^{8} - c x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________