Optimal. Leaf size=104 \[ -\frac {2 \text {ArcTan}\left (\frac {(-1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{3 (-1+k)}-\frac {4 \text {ArcTan}\left (\frac {\sqrt {1+k+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{3 \sqrt {1+k+k^2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.63, antiderivative size = 362, normalized size of antiderivative = 3.48, number of steps
used = 19, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6850, 6857,
116, 6820, 948, 12, 174, 551} \begin {gather*} \frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{k};\text {ArcSin}\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};\text {ArcSin}\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};\text {ArcSin}\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 (\text {ArcSin}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 116
Rule 174
Rule 551
Rule 948
Rule 6820
Rule 6850
Rule 6857
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 \left (\frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}}+\frac {2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \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-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 \frac {1}{\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 {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}{\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 {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 \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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 22.32, size = 310, normalized size = 2.98 \begin {gather*} -\frac {2 \sqrt {1+\frac {1}{-1+x}} \sqrt {1+\frac {1-\frac {1}{k^2}}{-1+x}} (-1+x)^{3/2} \sqrt {-1+k^2 x} \left (-3 i \sqrt {3} \left (1+k^3\right ) F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )+2 i \sqrt {3} \left (1+k+k^2\right ) \Pi \left (\frac {-1+k}{k};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )+(-1+k) \left (\left (-2 i \sqrt {3}+\left (3-i \sqrt {3}\right ) k\right ) \Pi \left (\frac {2 \left (1+k+k^2\right )}{k \left (1-i \sqrt {3}+2 k\right )};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )-i \left (2 \sqrt {3}+\left (-3 i+\sqrt {3}\right ) k\right ) \Pi \left (\frac {2 \left (1+k+k^2\right )}{k \left (1+i \sqrt {3}+2 k\right )};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )\right )\right )}{3 \left (-1+k^3\right ) \sqrt {(-1+x) x \left (-1+k^2 x\right )} \sqrt {-3+3 k^2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.14, size = 363, normalized size = 3.49
method | result | size |
default | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {4 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{3 k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}-\frac {4 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (k^{2} \textit {\_Z}^{2}+k \textit {\_Z} +1\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha k -2\right ) \left (\underline {\hspace {1.25 ex}}\alpha \,k^{2}+k +1\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+k +1}{k^{2}+k +1}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha k +1\right ) \left (k^{2}+k +1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}\right )}{3 k}\) | \(363\) |
elliptic | \(\text {Expression too large to display}\) | \(872\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (90) = 180\).
time = 0.46, size = 207, normalized size = 1.99 \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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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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