Optimal. Leaf size=108 \[ -\frac {2 \tan ^{-1}\left (\frac {(k+1) x}{\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}\right )}{3 (k+1)}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {k^2-k+1} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{(x-1) \left (k^2 x-1\right )}\right )}{3 \sqrt {k^2-k+1}} \]
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Rubi [C] time = 2.56, antiderivative size = 364, normalized size of antiderivative = 3.37, number of steps used = 20, number of rules used = 9, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6718, 6688, 6725, 714, 115, 934, 12, 168, 537} \begin {gather*} \frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {1}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt [3]{-1}}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {(-1)^{2/3}}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 12
Rule 115
Rule 168
Rule 537
Rule 714
Rule 934
Rule 6688
Rule 6718
Rule 6725
Rubi steps
\begin {align*} \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^3 x^3}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1+k^3 x^3\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^3 x^3}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1+k^3 x^3\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {2}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1+k^3 x^3\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1+k^3 x^3\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (-\frac {1}{3 (-1-k x) \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {1}{3 \left (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {1}{3 \left (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{(-1-k x) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\left (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\left (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {2} \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} (-1-k x) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \sqrt {2} \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \left (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \sqrt {2} \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \left (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} (-1-k x) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \left (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \left (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (-1+k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (4 \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (-1-\sqrt [3]{-1} k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (4 \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (-1+(-1)^{2/3} k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {1}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt [3]{-1}}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {(-1)^{2/3}}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ \end {align*}
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Mathematica [C] time = 7.66, size = 314, normalized size = 2.91 \begin {gather*} -\frac {2 \sqrt {\frac {1}{x-1}+1} (x-1)^{3/2} \sqrt {\frac {1-\frac {1}{k^2}}{x-1}+1} \sqrt {k^2 x-1} \left (-2 i \sqrt {3} \left (k^2-k+1\right ) \Pi \left (1+\frac {1}{k};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+(k+1) \left (\left (\left (3+i \sqrt {3}\right ) k-2 i \sqrt {3}\right ) \Pi \left (\frac {2 \left (k^2-k+1\right )}{k \left (2 k-i \sqrt {3}-1\right )};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+i \left (-2 \sqrt {3}+\left (\sqrt {3}+3 i\right ) k\right ) \Pi \left (\frac {2 \left (k^2-k+1\right )}{k \left (2 k+i \sqrt {3}-1\right )};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )-3 i \sqrt {3} \left (k^3-1\right ) F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{3 \left (k^3+1\right ) \sqrt {(x-1) x \left (k^2 x-1\right )} \sqrt {3 k^2 x-3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 108, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {(k+1) x}{\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}\right )}{3 (k+1)}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {k^2-k+1} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{(x-1) \left (k^2 x-1\right )}\right )}{3 \sqrt {k^2-k+1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 217, normalized size = 2.01 \begin {gather*} \frac {2 \, \sqrt {k^{2} - k + 1} {\left (k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - {\left (2 \, k^{2} - k + 2\right )} x + 1\right )} \sqrt {k^{2} - k + 1}}{2 \, {\left ({\left (k^{4} - k^{3} + k^{2}\right )} x^{3} - {\left (k^{4} - k^{3} + 2 \, k^{2} - k + 1\right )} x^{2} + {\left (k^{2} - k + 1\right )} x\right )}}\right ) + {\left (k^{2} - k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right )}{3 \, {\left (k^{3} + 1\right )}} \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^{3} - 1}{{\left (k^{3} x^{3} + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 368, normalized size = 3.41 \begin {gather*} -\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {4 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (k^{2} \textit {\_Z}^{2}-k \textit {\_Z} +1\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha k +2\right ) \left (\underline {\hspace {1.25 ex}}\alpha \,k^{2}-k +1\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}-k +1}{k^{2}-k +1}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha k -1\right ) \left (k^{2}-k +1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}\right )}{3 k}+\frac {4 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{3 k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k}\right )} \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^{3} - 1}{{\left (k^{3} x^{3} + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \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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