Optimal. Leaf size=285 \[ -\frac {(-1)^{2/3} \log \left (k^2 x^2+2 (-1)^{2/3} \sqrt [3]{2} (k-1)^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1} k x\right ) \sqrt [3]{k x^3+(-k-1) x^2+x}-2 k x+1\right )}{2\ 2^{2/3} \sqrt [3]{k-1}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1} \sqrt [3]{k x^3+(-k-1) x^2+x}+k x-1\right )}{2^{2/3} \sqrt [3]{k-1}}+\frac {(-1)^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} k x-\sqrt {3}}{-2 \sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1} \sqrt [3]{k x^3+(-k-1) x^2+x}+k x-1}\right )}{2^{2/3} \sqrt [3]{k-1}} \]
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Rubi [F] time = 0.50, 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 {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-k x}}{(1-x)^{2/3} x^{2/3} (1+(-2+k) x)} \, dx}{((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.21, size = 285, normalized size = 1.00 \begin {gather*} -\frac {(-1)^{2/3} \log \left (k^2 x^2+2 (-1)^{2/3} \sqrt [3]{2} (k-1)^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1} k x\right ) \sqrt [3]{k x^3+(-k-1) x^2+x}-2 k x+1\right )}{2\ 2^{2/3} \sqrt [3]{k-1}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1} \sqrt [3]{k x^3+(-k-1) x^2+x}+k x-1\right )}{2^{2/3} \sqrt [3]{k-1}}+\frac {(-1)^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} k x-\sqrt {3}}{-2 \sqrt [3]{-1} 2^{2/3} \sqrt [3]{k-1} \sqrt [3]{k x^3+(-k-1) x^2+x}+k x-1}\right )}{2^{2/3} \sqrt [3]{k-1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 94.95, size = 932, normalized size = 3.27
result too large to display
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 x - 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (k - 2\right )} x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-k x +1}{\left (1+\left (-2+k \right ) x \right ) \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}}}\, 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 x - 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (k - 2\right )} x + 1\right )}}\,{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 {k\,x-1}{\left (x\,\left (k-2\right )+1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {k x}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {1}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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