Optimal. Leaf size=38 \[ \frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}+\frac {\sin ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6161, 50, 53, 619, 216} \[ \frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}+\frac {\sin ^{-1}(a+b x)}{b} \]
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 = 26, normalized size = 0.68 \[ \frac {\sqrt {1-(a+b x)^2}+\sin ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 77, normalized size = 2.03 \[ \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} - \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 44, normalized size = 1.16 \[ -\frac {\arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{{\left | b \right |}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 95, normalized size = 2.50 \[ \frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 37, normalized size = 0.97 \[ \frac {\arcsin \left (b x + a\right )}{b} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{a+b\,x+1} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{a + b x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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