Optimal. Leaf size=203 \[ -\frac {\log \left (d^{2/3} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}+\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (\sqrt [3]{d}-\sqrt [3]{d} k x\right )+k^2 x^2-2 k x+1\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x-1\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} k x-\sqrt {3}}{-2 \sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x-1}\right )}{\sqrt [3]{d}} \]
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Rubi [F] time = 8.05, 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 {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{\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 ) \int \left (-\frac {k \left (5 d+k^2\right )}{d^2 \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}}+\frac {k^2 x}{d \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}}+\frac {-k \left (8 d^2-k^2-d \left (5-k^2\right )\right )-\left (2 k^4-2 d^2 \left (1-k^2\right )+d k^2 \left (11+k^2\right )\right ) x+k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) x^2}{d^2 \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )}\right ) \, dx}{\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 ) \int \frac {-k \left (8 d^2-k^2-d \left (5-k^2\right )\right )-\left (2 k^4-2 d^2 \left (1-k^2\right )+d k^2 \left (11+k^2\right )\right ) x+k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (k^2 \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}} \, dx}{d \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}-\frac {\left (k \left (5 d+k^2\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=-\frac {k \left (5 d+k^2\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{d^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 ) \int \left (\frac {k \left (-8 d^2+k^2+d \left (5-k^2\right )\right )}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )}+\frac {\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )}+\frac {k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )}\right ) \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (k^2 \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} \left (1-k^2 x\right )^{2/3}} \, dx,x,x^2\right )}{2 d \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=-\frac {k \left (5 d+k^2\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {k^2 \operatorname {Subst}\left (\int \frac {1}{\left (1+\left (-1-k^2\right ) x+k^2 x^2\right )^{2/3}} \, dx,x,x^2\right )}{2 d}-\frac {\left (k \left (8 d^2-k^2-d \left (5-k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=-\frac {k \left (5 d+k^2\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}-\frac {\left (k \left (8 d^2-k^2-d \left (5-k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (3 k^2 \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 k^2+\left (-1-k^2\right )^2+4 k^2 x^3}} \, dx,x,\sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}\right )}{2 d \left (-1-k^2+2 k^2 x^2\right )}\\ &=-\frac {k \left (5 d+k^2\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} k^{4/3} \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2} \left (\left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right ) \sqrt {\frac {\left (-1+k^2\right )^{4/3}-2^{2/3} k^{2/3} \left (-1+k^2\right )^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}+2 \sqrt [3]{2} k^{4/3} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} d \left (1+k^2-2 k^2 x^2\right ) \sqrt {\left (-1-k^2 \left (1-2 x^2\right )\right )^2} \sqrt {\frac {\left (-1+k^2\right )^{2/3} \left (\left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )}{\left (\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}}}-\frac {\left (k \left (8 d^2-k^2-d \left (5-k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx}{d^2 \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 = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 8.12, size = 203, normalized size = 1.00 \begin {gather*} -\frac {\log \left (d^{2/3} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}+\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (\sqrt [3]{d}-\sqrt [3]{d} k x\right )+k^2 x^2-2 k x+1\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x-1\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} k x-\sqrt {3}}{-2 \sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x-1}\right )}{\sqrt [3]{d}} \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^{3} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (d k x^{3} - {\left (d + 2\right )} k x + {\left (k^{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 = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 k +2 \left (k^{2}+1\right ) x +k \left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}+k^{3} x^{4}}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {2}{3}} \left (1-d -\left (2+d \right ) k x +\left (k^{2}+d \right ) x^{2}+d k \,x^{3}\right )}\, dx \end {gather*}
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^{3} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (d k x^{3} - {\left (d + 2\right )} k x + {\left (k^{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 {2\,x\,\left (k^2+1\right )-3\,k-4\,k^2\,x^3+k^3\,x^4+k\,x^2\,\left (k^2+1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (d\,k\,x^3+\left (k^2+d\right )\,x^2-k\,\left (d+2\right )\,x-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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