Optimal. Leaf size=57 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {a+b} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x)}\right )}{\sqrt {a} \sqrt {a+b}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 6.06, antiderivative size = 338, normalized size of antiderivative = 5.93, number of steps
used = 12, number of rules used = 6, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6850, 6860,
116, 174, 552, 551} \begin {gather*} \frac {2 \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \Pi \left (-\frac {2 (a+b) k^2}{-2 a k^2-2 b k^2+b+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}};\text {ArcSin}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \Pi \left (\frac {2 (a+b) k^2}{2 a k^2-b \left (1-2 k^2\right )+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}};\text {ArcSin}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{(a+b) \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 )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 116
Rule 174
Rule 551
Rule 552
Rule 6850
Rule 6860
Rubi steps
\begin {align*} \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\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}{(a+b) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}}+\frac {2 a+b+\left (b-2 a k^2-2 b k^2\right ) x}{(a+b) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\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}{(a+b) \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 {2 a+b+\left (b-2 a k^2-2 b k^2\right ) x}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx}{(a+b) \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 )}{(a+b) \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 {b-2 a k^2-2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-b-\sqrt {b^2+4 a^2 k^2+4 a b k^2}+2 \left (a k^2+b k^2\right ) x\right )}+\frac {b-2 a k^2-2 b k^2-\sqrt {b^2+4 a^2 k^2+4 a b k^2}}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-b+\sqrt {b^2+4 a^2 k^2+4 a b k^2}+2 \left (a k^2+b k^2\right ) x\right )}\right ) \, dx}{(a+b) \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 )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2-2 b k^2-\sqrt {b^2+4 a^2 k^2+4 a b k^2}\right ) \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 (-b+\sqrt {b^2+4 a^2 k^2+4 a b k^2}+2 \left (a k^2+b k^2\right ) x\right )} \, dx}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2-2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}\right ) \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 (-b-\sqrt {b^2+4 a^2 k^2+4 a b k^2}+2 \left (a k^2+b k^2\right ) x\right )} \, dx}{(a+b) \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 )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (2 \left (b-2 a k^2-2 b k^2-\sqrt {b^2+4 a^2 k^2+4 a b k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2+k^2 x^2} \left (2 a k^2-b \left (1-2 k^2\right )+\sqrt {b^2+4 a^2 k^2+4 a b k^2}-2 (a+b) k^2 x^2\right )} \, dx,x,\sqrt {1-x}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (2 \left (b-2 a k^2-2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2+k^2 x^2} \left (2 a k^2-b \left (1-2 k^2\right )-\sqrt {b^2+4 a^2 k^2+4 a b k^2}-2 (a+b) k^2 x^2\right )} \, dx,x,\sqrt {1-x}\right )}{(a+b) \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 )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (2 \left (b-2 a k^2-2 b k^2-\sqrt {b^2+4 a^2 k^2+4 a b k^2}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (2 a k^2-b \left (1-2 k^2\right )+\sqrt {b^2+4 a^2 k^2+4 a b k^2}-2 (a+b) k^2 x^2\right ) \sqrt {1+\frac {k^2 x^2}{1-k^2}}} \, dx,x,\sqrt {1-x}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (2 \left (b-2 a k^2-2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (2 a k^2-b \left (1-2 k^2\right )-\sqrt {b^2+4 a^2 k^2+4 a b k^2}-2 (a+b) k^2 x^2\right ) \sqrt {1+\frac {k^2 x^2}{1-k^2}}} \, dx,x,\sqrt {1-x}\right )}{(a+b) \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 )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \Pi \left (-\frac {2 (a+b) k^2}{b-2 a k^2-2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \Pi \left (\frac {2 (a+b) k^2}{2 a k^2-b \left (1-2 k^2\right )+\sqrt {b^2+4 a^2 k^2+4 a b k^2}};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 21.64, size = 227, normalized size = 3.98 \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 (F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )-\Pi \left (-\frac {2 (a+b) \left (-1+k^2\right )}{b-2 a k^2-2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )-\Pi \left (\frac {2 (a+b) \left (-1+k^2\right )}{-b+2 a k^2+2 b k^2+\sqrt {b^2+4 a^2 k^2+4 a b k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|1-\frac {1}{k^2}\right )\right )}{(a+b) \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 = 4588, normalized size = 80.49
method | result | size |
default | \(\text {Expression too large to display}\) | \(4588\) |
elliptic | \(\text {Expression too large to display}\) | \(4605\) |
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. 112 vs.
\(2 (46) = 92\).
time = 0.66, size = 345, normalized size = 6.05 \begin {gather*} \left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} k^{2} x^{3} + {\left (6 \, {\left (a^{2} + a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} x}{{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (a b + b^{2}\right )} k^{2} x^{3} + 2 \, a b x - {\left (2 \, {\left (a^{2} + a b\right )} k^{2} - b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} + a b}}{2 \, {\left ({\left (a^{2} + a b\right )} k^{2} x^{3} - {\left ({\left (a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} + a b}}\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 [B]
time = 3.84, size = 122, normalized size = 2.14 \begin {gather*} \frac {\ln \left (\frac {a\,\sqrt {a\,\left (a+b\right )}-2\,a\,x\,\sqrt {a\,\left (a+b\right )}-b\,x\,\sqrt {a\,\left (a+b\right )}+a\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+b\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+a\,\left (a+b\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a+b\,x-a\,k^2\,x^2-b\,k^2\,x^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2+b\,a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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