Optimal. Leaf size=39 \[ \frac {\sin ^{-1}(a+b x)}{b}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6161, 50, 53, 619, 216} \[ \frac {\sin ^{-1}(a+b x)}{b}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 216
Rule 619
Rule 6161
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a+b x)} \, dx &=\int \frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{b}+\int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{b}+\int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{2 b^2}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{b}+\frac {\sin ^{-1}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.72 \[ \frac {\sin ^{-1}(a+b x)-\sqrt {1-(a+b x)^2}}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 76, normalized size = 1.95 \[ -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} + \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 36, normalized size = 0.92 \[ -\frac {\arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{{\left | b \right |}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 71, normalized size = 1.82 \[ -\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 65, normalized size = 1.67 \[ -\frac {\arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 101, normalized size = 2.59 \[ \frac {\mathrm {asin}\left (a+b\,x\right )}{b}-\frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{b}-\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{\sqrt {-b^2}}+\frac {a\,\mathrm {asin}\left (a+b\,x\right )}{b} \]
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 \[ \int \frac {a + b x + 1}{\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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