Optimal. Leaf size=55 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {-2+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{\sqrt {-2+k^2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 1.97, antiderivative size = 508, normalized size of antiderivative = 9.24, number of steps
used = 17, number of rules used = 9, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.155, Rules used = {6, 2081, 6860,
730, 1117, 948, 174, 552, 551} \begin {gather*} -\frac {2 \sqrt {-k^4+2 k^2-1} \left (k^2+2 \sqrt {k^2-1}\right ) \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \sqrt {k^2 x-k^2} \Pi \left (-\frac {k^4-3 k^2+2}{k^2 \left (-k^2-\sqrt {k^2-1}+1\right )};\text {ArcSin}\left (\frac {\sqrt {k^2 x-k^2}}{\sqrt {1-k^2}}\right )|1-\frac {1}{k^2}\right )}{\left (2-k^2\right ) \left (-k^2-\sqrt {k^2-1}+1\right ) k^2 \sqrt {k^2 x^3-\left (k^2+1\right ) x^2+x}}+\frac {2 \sqrt {-k^4+2 k^2-1} \left (k^2-2 \sqrt {k^2-1}\right ) \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \sqrt {k^2 x-k^2} \Pi \left (-\frac {k^4-3 k^2+2}{k^2 \left (-k^2+\sqrt {k^2-1}+1\right )};\text {ArcSin}\left (\frac {\sqrt {k^2 x-k^2}}{\sqrt {1-k^2}}\right )|1-\frac {1}{k^2}\right )}{\left (2-k^2\right ) \left (-k^2+\sqrt {k^2-1}+1\right ) k^2 \sqrt {k^2 x^3-\left (k^2+1\right ) x^2+x}}-\frac {k^{3/2} \sqrt {x} (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} \sqrt {x}\right )|\frac {(k+1)^2}{4 k}\right )}{\left (2-k^2\right ) \sqrt {k^2 x^3-\left (k^2+1\right ) x^2+x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 6
Rule 174
Rule 551
Rule 552
Rule 730
Rule 948
Rule 1117
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {1-2 x+k^2 x^2}{\left (-1+2 x-2 x^2+k^2 x^2\right ) \sqrt {x-x^2-k^2 x^2+k^2 x^3}} \, dx &=\int \frac {1-2 x+k^2 x^2}{\left (-1+2 x+\left (-2+k^2\right ) x^2\right ) \sqrt {x-x^2-k^2 x^2+k^2 x^3}} \, dx\\ &=\int \frac {1-2 x+k^2 x^2}{\left (-1+2 x+\left (-2+k^2\right ) x^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \frac {1-2 x+k^2 x^2}{\sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \left (-\frac {k^2}{\left (2-k^2\right ) \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}}+\frac {2 \left (-1+k^2\right ) (1-2 x)}{\left (-2+k^2\right ) \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )}\right ) \, dx}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {\left (k^2 \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}+\frac {\left (2 \left (1-k^2\right ) \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \frac {1-2 x}{\sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {\left (2 k^2 \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx,x,\sqrt {x}\right )}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}+\frac {\left (2 \left (1-k^2\right ) \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \left (\frac {-2+\frac {k^2}{\sqrt {-1+k^2}}}{\sqrt {x} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}}+\frac {-2-\frac {k^2}{\sqrt {-1+k^2}}}{\sqrt {x} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}}\right ) \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {k^{3/2} \sqrt {x} (1+k x) \sqrt {\frac {1-\left (1+k^2\right ) x+k^2 x^2}{(1+k x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {k} \sqrt {x}\right )|\frac {(1+k)^2}{4 k}\right )}{\left (2-k^2\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}+\frac {\left (2 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \frac {1}{\sqrt {x} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}+\frac {\left (2 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {x} \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}\right ) \int \frac {1}{\sqrt {x} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {k^{3/2} \sqrt {x} (1+k x) \sqrt {\frac {1-\left (1+k^2\right ) x+k^2 x^2}{(1+k x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {k} \sqrt {x}\right )|\frac {(1+k)^2}{4 k}\right )}{\left (2-k^2\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}+\frac {\left (2 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )} \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}+\frac {\left (2 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )} \, dx}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {k^{3/2} \sqrt {x} (1+k x) \sqrt {\frac {1-\left (1+k^2\right ) x+k^2 x^2}{(1+k x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {k} \sqrt {x}\right )|\frac {(1+k)^2}{4 k}\right )}{\left (2-k^2\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}-\frac {\left (4 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-2 \left (1-k^2\right )+x^2} \sqrt {1+\frac {x^2}{2 k^2}} \left (4 k^2 \left (1-k^2-\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {-2 k^2+2 k^2 x}\right )}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}-\frac {\left (4 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-2 \left (1-k^2\right )+x^2} \sqrt {1+\frac {x^2}{2 k^2}} \left (4 k^2 \left (1-k^2+\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {-2 k^2+2 k^2 x}\right )}{\left (2-k^2\right ) \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {k^{3/2} \sqrt {x} (1+k x) \sqrt {\frac {1-\left (1+k^2\right ) x+k^2 x^2}{(1+k x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {k} \sqrt {x}\right )|\frac {(1+k)^2}{4 k}\right )}{\left (2-k^2\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}-\frac {\left (2 \sqrt {2} \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2 k^2}} \sqrt {1-\frac {x^2}{2 \left (1-k^2\right )}} \left (4 k^2 \left (1-k^2-\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {-2 k^2+2 k^2 x}\right )}{\left (2-k^2\right ) \sqrt {-1+k^2 x} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}-\frac {\left (2 \sqrt {2} \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {x} \sqrt {-2+2 k^2 x} \sqrt {-2 k^2+2 k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2 k^2}} \sqrt {1-\frac {x^2}{2 \left (1-k^2\right )}} \left (4 k^2 \left (1-k^2+\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {-2 k^2+2 k^2 x}\right )}{\left (2-k^2\right ) \sqrt {-1+k^2 x} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\\ &=-\frac {k^{3/2} \sqrt {x} (1+k x) \sqrt {\frac {1-\left (1+k^2\right ) x+k^2 x^2}{(1+k x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {k} \sqrt {x}\right )|\frac {(1+k)^2}{4 k}\right )}{\left (2-k^2\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}-\frac {2 \sqrt {-\left (1-k^2\right )^2} \left (k^2+2 \sqrt {-1+k^2}\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {x} \sqrt {-k^2+k^2 x} \Pi \left (-\frac {2-3 k^2+k^4}{k^2 \left (1-k^2-\sqrt {-1+k^2}\right )};\sin ^{-1}\left (\frac {\sqrt {-k^2+k^2 x}}{\sqrt {1-k^2}}\right )|1-\frac {1}{k^2}\right )}{k^2 \left (2-k^2\right ) \left (1-k^2-\sqrt {-1+k^2}\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}+\frac {2 \left (1-k^2\right )^{3/2} \left (2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {x} \sqrt {-k^2+k^2 x} \Pi \left (-\frac {2-3 k^2+k^4}{k^2 \left (1-k^2+\sqrt {-1+k^2}\right )};\sin ^{-1}\left (\frac {\sqrt {-k^2+k^2 x}}{\sqrt {1-k^2}}\right )|1-\frac {1}{k^2}\right )}{k^2 \left (2-k^2\right ) \left (1-k^2+\sqrt {-1+k^2}\right ) \sqrt {x-\left (1+k^2\right ) x^2+k^2 x^3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 10.68, size = 202, normalized size = 3.67 \begin {gather*} \frac {2 i \sqrt {1+\frac {1}{-1+x}} \sqrt {1+\frac {1-\frac {1}{k^2}}{-1+x}} (-1+x)^{3/2} \left (\left (-2+k^2\right ) F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )+\left (1+\sqrt {-1+k^2}\right ) \Pi \left (\frac {-1+k^2}{-1+k^2-\sqrt {-1+k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )-\left (-1+\sqrt {-1+k^2}\right ) \Pi \left (\frac {-1+k^2}{-1+k^2+\sqrt {-1+k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )\right )}{\left (-2+k^2\right ) \sqrt {(-1+x) x \left (-1+k^2 x\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.84, size = 2705, normalized size = 49.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(2705\) |
elliptic | \(\text {Expression too large to display}\) | \(2727\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 269, normalized size = 4.89 \begin {gather*} \left [-\frac {\sqrt {-k^{2} + 2} \log \left (\frac {{\left (k^{4} - 4 \, k^{2} + 4\right )} x^{4} - 4 \, {\left (2 \, k^{4} - 5 \, k^{2} + 2\right )} x^{3} + 2 \, {\left (4 \, k^{4} - 5 \, k^{2} - 4\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left ({\left (k^{2} - 2\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + 1\right )} \sqrt {-k^{2} + 2} - 4 \, {\left (2 \, k^{2} - 3\right )} x + 1}{{\left (k^{4} - 4 \, k^{2} + 4\right )} x^{4} + 4 \, {\left (k^{2} - 2\right )} x^{3} - 2 \, {\left (k^{2} - 4\right )} x^{2} - 4 \, x + 1}\right )}{2 \, {\left (k^{2} - 2\right )}}, \frac {\arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left ({\left (k^{2} - 2\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + 1\right )} \sqrt {k^{2} - 2}}{2 \, {\left ({\left (k^{4} - 2 \, k^{2}\right )} x^{3} - {\left (k^{4} - k^{2} - 2\right )} x^{2} + {\left (k^{2} - 2\right )} x\right )}}\right )}{\sqrt {k^{2} - 2}}\right ] \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^{2} x^{2} - 2 x + 1}{\sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k^{2} x^{2} - 2 x^{2} + 2 x - 1\right )}\, dx \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 [B]
time = 1.83, size = 84, normalized size = 1.53 \begin {gather*} \frac {\ln \left (\frac {x\,2{}\mathrm {i}+k^2\,x^2\,1{}\mathrm {i}-k^2\,x\,2{}\mathrm {i}-x^2\,2{}\mathrm {i}-2\,\sqrt {k^2-2}\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}+1{}\mathrm {i}}{2\,k^2\,x^2-4\,x^2+4\,x-2}\right )\,1{}\mathrm {i}}{\sqrt {k^2-2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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