Optimal. Leaf size=256 \[ \frac {\log \left (d^{2/3} k^4 x^4-2 d^{2/3} k^2 x^2+d^{2/3}+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (\sqrt [3]{d}-\sqrt [3]{d} k^2 x^2\right )+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{4/3}\right )}{4 \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} k^2 x^2-\sqrt [3]{d}+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}\right )}{2 \sqrt [3]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}{-2 \sqrt [3]{d} k^2 x^2+2 \sqrt [3]{d}+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}\right )}{2 \sqrt [3]{d}} \]
________________________________________________________________________________________
Rubi [C] time = 2.57, antiderivative size = 275, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 7, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {1594, 6715, 6719, 1586, 6728, 137, 136} \begin {gather*} \frac {3 \left (1-x^2\right )^2 \left (\frac {1-k^2 x^2}{1-k^2}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {k^2 \left (1-x^2\right )}{1-k^2},\frac {2 \left (1-x^2\right )}{\sqrt {d} \left (\sqrt {d} k^2-\sqrt {d k^4-4 k^2+4}\right )}\right )}{8 d \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {3 \left (1-x^2\right )^2 \left (\frac {1-k^2 x^2}{1-k^2}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {k^2 \left (1-x^2\right )}{1-k^2},\frac {2 \left (1-x^2\right )}{\sqrt {d} \left (\sqrt {d} k^2+\sqrt {d k^4-4 k^2+4}\right )}\right )}{8 d \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 136
Rule 137
Rule 1586
Rule 1594
Rule 6715
Rule 6719
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx &=\int \frac {x \left (2-k^2-2 x^2+k^2 x^4\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2-k^2-2 x+k^2 x^2}{\left ((1-x) \left (1-k^2 x\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {2-k^2-2 x+k^2 x^2}{(1-x)^{2/3} \left (1-k^2 x\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x+x^2\right )} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x} \left (2-k^2-k^2 x\right )}{\left (1-k^2 x\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x+x^2\right )} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (-k^2+\frac {\sqrt {4-4 k^2+d k^4}}{\sqrt {d}}\right ) \sqrt [3]{1-x}}{\left (-2+d k^2-\sqrt {d} \sqrt {4-4 k^2+d k^4}+2 x\right ) \left (1-k^2 x\right )^{2/3}}+\frac {\left (-k^2-\frac {\sqrt {4-4 k^2+d k^4}}{\sqrt {d}}\right ) \sqrt [3]{1-x}}{\left (-2+d k^2+\sqrt {d} \sqrt {4-4 k^2+d k^4}+2 x\right ) \left (1-k^2 x\right )^{2/3}}\right ) \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (-k^2-\frac {\sqrt {4-4 k^2+d k^4}}{\sqrt {d}}\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x}}{\left (-2+d k^2+\sqrt {d} \sqrt {4-4 k^2+d k^4}+2 x\right ) \left (1-k^2 x\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-k^2+\frac {\sqrt {4-4 k^2+d k^4}}{\sqrt {d}}\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x}}{\left (-2+d k^2-\sqrt {d} \sqrt {4-4 k^2+d k^4}+2 x\right ) \left (1-k^2 x\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (-k^2-\frac {\sqrt {4-4 k^2+d k^4}}{\sqrt {d}}\right ) \left (1-x^2\right )^{2/3} \left (\frac {-1+k^2 x^2}{-1+k^2}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x}}{\left (-2+d k^2+\sqrt {d} \sqrt {4-4 k^2+d k^4}+2 x\right ) \left (-\frac {1}{-1+k^2}+\frac {k^2 x}{-1+k^2}\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-k^2+\frac {\sqrt {4-4 k^2+d k^4}}{\sqrt {d}}\right ) \left (1-x^2\right )^{2/3} \left (\frac {-1+k^2 x^2}{-1+k^2}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x}}{\left (-2+d k^2-\sqrt {d} \sqrt {4-4 k^2+d k^4}+2 x\right ) \left (-\frac {1}{-1+k^2}+\frac {k^2 x}{-1+k^2}\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {3 \left (1-x^2\right )^2 \left (\frac {1-k^2 x^2}{1-k^2}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {k^2 \left (1-x^2\right )}{1-k^2},\frac {2 \left (1-x^2\right )}{\sqrt {d} \left (\sqrt {d} k^2-\sqrt {4-4 k^2+d k^4}\right )}\right )}{8 d \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {3 \left (1-x^2\right )^2 \left (\frac {1-k^2 x^2}{1-k^2}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {k^2 \left (1-x^2\right )}{1-k^2},\frac {2 \left (1-x^2\right )}{\sqrt {d} \left (\sqrt {d} k^2+\sqrt {4-4 k^2+d k^4}\right )}\right )}{8 d \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 2.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 4.70, size = 256, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} k^2 x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} k^2 x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} k^2 x^2\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{4/3}\right )}{4 \sqrt [3]{d}} \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 {k^{2} x^{5} - 2 \, x^{3} - {\left (k^{2} - 2\right )} x}{{\left (x^{4} + {\left (d k^{2} - 2\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-k^{2}+2\right ) x -2 x^{3}+k^{2} x^{5}}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {2}{3}} \left (1-d +\left (d \,k^{2}-2\right ) x^{2}+x^{4}\right )}\, dx\]
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 {k^{2} x^{5} - 2 \, x^{3} - {\left (k^{2} - 2\right )} x}{{\left (x^{4} + {\left (d k^{2} - 2\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}}\,{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 = -0.00 \begin {gather*} -\int \frac {x\,\left (k^2-2\right )-k^2\,x^5+2\,x^3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (x^4+\left (d\,k^2-2\right )\,x^2-d+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________