Optimal. Leaf size=66 \[ -\sqrt {2} \sqrt {\sqrt {a^2+1}+a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+1}-a} (x-a)}{\sqrt {\left (x^2+1\right ) (x-a)}}\right ) \]
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Rubi [C] time = 1.23, antiderivative size = 204, normalized size of antiderivative = 3.09, number of steps used = 9, number of rules used = 8, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6719, 6742, 719, 419, 932, 168, 538, 537} \[ \frac {4 \sqrt {a^2+1} \sqrt {x^2+1} \sqrt {\frac {a-x}{a+i}} \Pi \left (\frac {2}{1-i \left (a-\sqrt {a^2+1}\right )};\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2}{1-i a}\right )}{\left (1-i \left (a-\sqrt {a^2+1}\right )\right ) \sqrt {\left (x^2+1\right ) (-(a-x))}}+\frac {2 i \sqrt {x^2+1} \sqrt {\frac {a-x}{a+i}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2}{1-i a}\right )}{\sqrt {\left (x^2+1\right ) (-(a-x))}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 537
Rule 538
Rule 719
Rule 932
Rule 6719
Rule 6742
Rubi steps
\begin {align*} \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx &=\frac {\left (\sqrt {-a+x} \sqrt {1+x^2}\right ) \int \frac {-a-\sqrt {1+a^2}+x}{\sqrt {-a+x} \left (-a+\sqrt {1+a^2}+x\right ) \sqrt {1+x^2}} \, dx}{\sqrt {(-a+x) \left (1+x^2\right )}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {1+x^2}\right ) \int \left (\frac {1}{\sqrt {-a+x} \sqrt {1+x^2}}-\frac {2 \sqrt {1+a^2}}{\sqrt {-a+x} \left (-a+\sqrt {1+a^2}+x\right ) \sqrt {1+x^2}}\right ) \, dx}{\sqrt {(-a+x) \left (1+x^2\right )}}\\ &=\frac {\left (\sqrt {-a+x} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {-a+x} \sqrt {1+x^2}} \, dx}{\sqrt {(-a+x) \left (1+x^2\right )}}-\frac {\left (2 \sqrt {1+a^2} \sqrt {-a+x} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {-a+x} \left (-a+\sqrt {1+a^2}+x\right ) \sqrt {1+x^2}} \, dx}{\sqrt {(-a+x) \left (1+x^2\right )}}\\ &=-\frac {\left (2 \sqrt {1+a^2} \sqrt {-a+x} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-i x} \sqrt {1+i x} \sqrt {-a+x} \left (-a+\sqrt {1+a^2}+x\right )} \, dx}{\sqrt {(-a+x) \left (1+x^2\right )}}+\frac {\left (2 i \sqrt {\frac {-a+x}{-i-a}} \sqrt {1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 i x^2}{-i-a}}} \, dx,x,\frac {\sqrt {1-i x}}{\sqrt {2}}\right )}{\sqrt {(-a+x) \left (1+x^2\right )}}\\ &=\frac {2 i \sqrt {\frac {a-x}{i+a}} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2}{1-i a}\right )}{\sqrt {-(a-x) \left (1+x^2\right )}}+\frac {\left (4 \sqrt {1+a^2} \sqrt {-a+x} \sqrt {1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (1-i \left (a-\sqrt {1+a^2}\right )-x^2\right ) \sqrt {-i-a+i x^2}} \, dx,x,\sqrt {1-i x}\right )}{\sqrt {(-a+x) \left (1+x^2\right )}}\\ &=\frac {2 i \sqrt {\frac {a-x}{i+a}} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2}{1-i a}\right )}{\sqrt {-(a-x) \left (1+x^2\right )}}+\frac {\left (4 \sqrt {1+a^2} \sqrt {\frac {a-x}{i+a}} \sqrt {1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (1-i \left (a-\sqrt {1+a^2}\right )-x^2\right ) \sqrt {1+\frac {i x^2}{-i-a}}} \, dx,x,\sqrt {1-i x}\right )}{\sqrt {(-a+x) \left (1+x^2\right )}}\\ &=\frac {2 i \sqrt {\frac {a-x}{i+a}} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2}{1-i a}\right )}{\sqrt {-(a-x) \left (1+x^2\right )}}+\frac {4 \sqrt {1+a^2} \sqrt {\frac {a-x}{i+a}} \sqrt {1+x^2} \Pi \left (\frac {2}{1-i \left (a-\sqrt {1+a^2}\right )};\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2}{1-i a}\right )}{\left (1-i \left (a-\sqrt {1+a^2}\right )\right ) \sqrt {-(a-x) \left (1+x^2\right )}}\\ \end {align*}
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Mathematica [C] time = 1.15, size = 213, normalized size = 3.23 \[ \frac {2 \sqrt {\frac {a-x}{a+i}} \left (2 i \sqrt {a^2+1} \sqrt {1-i x} \sqrt {x^2+1} \Pi \left (\frac {2 i}{a-\sqrt {a^2+1}+i};\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right )|\frac {2 i}{a+i}\right )-\left (\sqrt {a^2+1}-a-i\right ) \sqrt {1+i x} (x+i) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {1-i x}}{\sqrt {2}}\right ),\frac {2 i}{a+i}\right )\right )}{\left (-\sqrt {a^2+1}+a+i\right ) \sqrt {1-i x} \sqrt {\left (x^2+1\right ) (x-a)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 546, normalized size = 8.27 \[ \left [\frac {1}{4} \, \sqrt {-2 \, a - 2 \, \sqrt {a^{2} + 1}} \log \left (-\frac {8 \, a x^{7} + x^{8} + 4 \, {\left (2 \, a^{2} + 15\right )} x^{6} - 8 \, {\left (4 \, a^{3} + 15 \, a\right )} x^{5} + 2 \, {\left (8 \, a^{4} + 80 \, a^{2} + 67\right )} x^{4} + 64 \, a^{4} - 8 \, {\left (20 \, a^{3} + 37 \, a\right )} x^{3} + 4 \, {\left (16 \, a^{4} + 74 \, a^{2} + 15\right )} x^{2} + 48 \, a^{2} - 4 \, {\left (a x^{6} + 2 \, {\left (2 \, a^{2} + 3\right )} x^{5} - {\left (4 \, a^{3} - a\right )} x^{4} - 8 \, a^{3} - {\left (4 \, a^{3} + 29 \, a\right )} x^{2} + 20 \, x^{3} + 2 \, {\left (10 \, a^{2} + 3\right )} x - {\left (4 \, a x^{5} + x^{6} - {\left (4 \, a^{2} - 15\right )} x^{4} - 16 \, a x^{3} + {\left (4 \, a^{2} + 15\right )} x^{2} + 8 \, a^{2} - 20 \, a x + 1\right )} \sqrt {a^{2} + 1} - 5 \, a\right )} \sqrt {-a x^{2} + x^{3} - a + x} \sqrt {-2 \, a - 2 \, \sqrt {a^{2} + 1}} - 8 \, {\left (24 \, a^{3} + 13 \, a\right )} x + 16 \, {\left (a x^{6} - x^{7} + 15 \, a x^{4} - 7 \, x^{5} - {\left (12 \, a^{2} + 7\right )} x^{3} + 4 \, a^{3} + {\left (4 \, a^{3} + 15 \, a\right )} x^{2} - {\left (12 \, a^{2} + 1\right )} x + a\right )} \sqrt {a^{2} + 1} + 1}{8 \, a x^{7} - x^{8} - 4 \, {\left (6 \, a^{2} - 1\right )} x^{6} + 8 \, {\left (4 \, a^{3} - 3 \, a\right )} x^{5} - 2 \, {\left (8 \, a^{4} - 24 \, a^{2} + 3\right )} x^{4} - 8 \, {\left (4 \, a^{3} - 3 \, a\right )} x^{3} - 4 \, {\left (6 \, a^{2} - 1\right )} x^{2} - 8 \, a x - 1}\right ), -\frac {1}{2} \, \sqrt {2 \, a + 2 \, \sqrt {a^{2} + 1}} \arctan \left (-\frac {\sqrt {-a x^{2} + x^{3} - a + x} {\left (2 \, a^{2} - 2 \, a x - x^{2} - 2 \, \sqrt {a^{2} + 1} {\left (a - x\right )} - 1\right )} \sqrt {2 \, a + 2 \, \sqrt {a^{2} + 1}}}{4 \, {\left (a x^{2} - x^{3} + a - x\right )}}\right )\right ] \]
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 \[ \int \frac {a - x + \sqrt {a^{2} + 1}}{\sqrt {-{\left (x^{2} + 1\right )} {\left (a - x\right )}} {\left (a - x - \sqrt {a^{2} + 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 1275, normalized size = 19.32 \[ \frac {2 \left (-a -i\right ) \sqrt {\frac {-a +x}{-a -i}}\, \sqrt {\frac {x -i}{a -i}}\, \sqrt {\frac {x +i}{a +i}}\, \EllipticF \left (\sqrt {\frac {-a +x}{-a -i}}, \sqrt {\frac {a +i}{a -i}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}}-2 \sqrt {a^{2}+1}\, \left (-\frac {i \sqrt {-i x +1}\, \sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {i x +1}\, a^{2} \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-a -i-\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-a -i}}\right )}{\sqrt {a^{2}+1}\, \sqrt {-a^{3} x^{2}+a^{2} x^{3}-a^{3}+a^{2} x -a \,x^{2}+x^{3}-a +x}\, \left (-a -i-\sqrt {a^{2}+1}\right )}+\frac {i \sqrt {-i x +1}\, \sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {i x +1}\, a^{2} \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-a -i+\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-a -i}}\right )}{\sqrt {a^{2}+1}\, \sqrt {-a^{3} x^{2}+a^{2} x^{3}-a^{3}+a^{2} x -a \,x^{2}+x^{3}-a +x}\, \left (-a -i+\sqrt {a^{2}+1}\right )}-\frac {\sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {\frac {x}{a -i}-\frac {i}{a -i}}\, \sqrt {\frac {x}{a +i}+\frac {i}{a +i}}\, a \EllipticPi \left (\sqrt {\frac {-a +x}{-a -i}}, \frac {a +i}{\sqrt {a^{2}+1}}, \sqrt {\frac {a +i}{a -i}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}\, \sqrt {a^{2}+1}}+\frac {\sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {\frac {x}{a -i}-\frac {i}{a -i}}\, \sqrt {\frac {x}{a +i}+\frac {i}{a +i}}\, a \EllipticPi \left (\sqrt {\frac {-a +x}{-a -i}}, -\frac {a +i}{\sqrt {a^{2}+1}}, \sqrt {\frac {a +i}{a -i}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}\, \sqrt {a^{2}+1}}-\frac {i \sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {\frac {x}{a -i}-\frac {i}{a -i}}\, \sqrt {\frac {x}{a +i}+\frac {i}{a +i}}\, \EllipticPi \left (\sqrt {\frac {-a +x}{-a -i}}, \frac {a +i}{\sqrt {a^{2}+1}}, \sqrt {\frac {a +i}{a -i}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}\, \sqrt {a^{2}+1}}+\frac {i \sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {\frac {x}{a -i}-\frac {i}{a -i}}\, \sqrt {\frac {x}{a +i}+\frac {i}{a +i}}\, \EllipticPi \left (\sqrt {\frac {-a +x}{-a -i}}, -\frac {a +i}{\sqrt {a^{2}+1}}, \sqrt {\frac {a +i}{a -i}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}\, \sqrt {a^{2}+1}}-\frac {i \sqrt {-i x +1}\, \sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {i x +1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-a -i-\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-a -i}}\right )}{\sqrt {a^{2}+1}\, \sqrt {-a^{3} x^{2}+a^{2} x^{3}-a^{3}+a^{2} x -a \,x^{2}+x^{3}-a +x}\, \left (-a -i-\sqrt {a^{2}+1}\right )}+\frac {i \sqrt {-i x +1}\, \sqrt {-\frac {a}{-a -i}+\frac {x}{-a -i}}\, \sqrt {i x +1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-a -i+\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-a -i}}\right )}{\sqrt {a^{2}+1}\, \sqrt {-a^{3} x^{2}+a^{2} x^{3}-a^{3}+a^{2} x -a \,x^{2}+x^{3}-a +x}\, \left (-a -i+\sqrt {a^{2}+1}\right )}\right ) \]
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 \[ \int \frac {a - x + \sqrt {a^{2} + 1}}{\sqrt {-{\left (x^{2} + 1\right )} {\left (a - x\right )}} {\left (a - x - \sqrt {a^{2} + 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int -\frac {a-x+\sqrt {a^2+1}}{\sqrt {-\left (x^2+1\right )\,\left (a-x\right )}\,\left (x-a+\sqrt {a^2+1}\right )} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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