Optimal. Leaf size=44 \[ \frac {(1-a) \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {\sin ^{-1}(a+b x)}{b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6164, 78, 53, 619, 216} \[ \frac {(1-a) \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {\sin ^{-1}(a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 53
Rule 78
Rule 216
Rule 619
Rule 6164
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a+b x)} x}{1-a^2-2 a b x-b^2 x^2} \, dx &=\int \frac {x}{(1-a-b x)^{3/2} \sqrt {1+a+b x}} \, dx\\ &=\frac {(1-a) \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}-\frac {\int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{b}\\ &=\frac {(1-a) \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}-\frac {\int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{b}\\ &=\frac {(1-a) \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\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^3}\\ &=\frac {(1-a) \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}-\frac {\sin ^{-1}(a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 49, normalized size = 1.11 \[ -\frac {\sin ^{-1}(a+b x)-\frac {(a-1) \sqrt {-a^2-2 a b x-b^2 x^2+1}}{a+b x-1}}{b^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.57, size = 98, normalized size = 2.23 \[ \frac {{\left (b x + a - 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a - 1\right )}}{b^{3} x + {\left (a - 1\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 66, normalized size = 1.50 \[ \frac {\arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{b {\left | b \right |}} - \frac {2 \, {\left (a - 1\right )}}{b {\left (\frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 160, normalized size = 3.64 \[ \frac {x}{b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}+\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a x}{b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 120, normalized size = 2.73 \[ \frac {b^{2} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{4} x + a b^{3} - b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4} x + a b^{3} - b^{3}} - \frac {\arcsin \left (b x + a\right )}{b^{3}}\right )}}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 229, normalized size = 5.20 \[ \frac {\left (\frac {a^2\,x}{b}+\frac {a\,\left (a^2-1\right )}{b^2}\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{a^2+2\,a\,b\,x+b^2\,x^2-1}+\frac {b\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{{\left (-b^2\right )}^{3/2}}+\frac {\left (\frac {a^2-1}{b^2}+\frac {a\,x}{b}\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{a^2+2\,a\,b\,x+b^2\,x^2-1}+\frac {x\,\left (b^2\,\left (a^2-1\right )-2\,a^2\,b^2\right )-a\,b\,\left (a^2-1\right )}{b\,\left (b^2\,\left (a^2-1\right )-a^2\,b^2\right )\,\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}} \]
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 \[ - \int \frac {x}{a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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