Optimal. Leaf size=243 \[ -\frac {\log \left (\sqrt [3]{d} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}+k^2 x^2-1\right )}{2 d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}{\sqrt [3]{d} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}-2 k^2 x^2+2}\right )}{2 d^{2/3}}+\frac {\log \left (d^{2/3} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{4/3}+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (\sqrt [3]{d}-\sqrt [3]{d} k^2 x^2\right )+k^4 x^4-2 k^2 x^2+1\right )}{4 d^{2/3}} \]
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Rubi [C] time = 2.44, antiderivative size = 251, normalized size of antiderivative = 1.03, 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 d \left (1-x^2\right )}{k^2-\sqrt {k^4-4 d k^2+4 d}}\right )}{8 \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 d \left (1-x^2\right )}{k^2+\sqrt {k^4-4 d k^2+4 d}}\right )}{8 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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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+d 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+d 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+d 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+d 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+d 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+\sqrt {4 d-4 d k^2+k^4}\right ) \sqrt [3]{1-x}}{\left (-2 d+k^2-\sqrt {4 d-4 d k^2+k^4}+2 d x\right ) \left (1-k^2 x\right )^{2/3}}+\frac {\left (-k^2-\sqrt {4 d-4 d k^2+k^4}\right ) \sqrt [3]{1-x}}{\left (-2 d+k^2+\sqrt {4 d-4 d k^2+k^4}+2 d 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-\sqrt {4 d-4 d k^2+k^4}\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 {4 d-4 d k^2+k^4}+2 d 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+\sqrt {4 d-4 d k^2+k^4}\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 {4 d-4 d k^2+k^4}+2 d 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-\sqrt {4 d-4 d k^2+k^4}\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 {4 d-4 d k^2+k^4}+2 d 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+\sqrt {4 d-4 d k^2+k^4}\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 {4 d-4 d k^2+k^4}+2 d 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 d \left (1-x^2\right )}{k^2-\sqrt {4 d-4 d k^2+k^4}}\right )}{8 \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 d \left (1-x^2\right )}{k^2+\sqrt {4 d-4 d k^2+k^4}}\right )}{8 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 1.88, 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+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.58, size = 243, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}{2-2 k^2 x^2+\sqrt [3]{d} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right )}{2 d^{2/3}}-\frac {\log \left (-1+k^2 x^2+\sqrt [3]{d} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}}+\frac {\log \left (1-2 k^2 x^2+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}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{4/3}\right )}{4 d^{2/3}} \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 {k^{2} x^{5} - 2 \, x^{3} - {\left (k^{2} - 2\right )} x}{{\left (d x^{4} + {\left (k^{2} - 2 \, d\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.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-k^{2}+2\right ) x -2 x^{3}+k^{2} x^{5}}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {2}{3}} \left (-1+d +\left (k^{2}-2 d \right ) x^{2}+d \,x^{4}\right )}\, dx\]
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 {k^{2} x^{5} - 2 \, x^{3} - {\left (k^{2} - 2\right )} x}{{\left (d x^{4} + {\left (k^{2} - 2 \, d\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.
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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 (d\,x^4+\left (k^2-2\,d\right )\,x^2+d-1\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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