Optimal. Leaf size=94 \[ \frac {2 b \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(a+1) \sqrt {1-a^2}}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(a+1) x} \]
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Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6163, 94, 93, 208} \[ \frac {2 b \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(a+1) \sqrt {1-a^2}}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(a+1) x} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 208
Rule 6163
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {\sqrt {1-a-b x}}{x^2 \sqrt {1+a+b x}} \, dx\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{(1+a) x}-\frac {b \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{1+a}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{(1+a) x}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{-1-a-(-1+a) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}}\right )}{1+a}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{(1+a) x}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1+a) \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 89, normalized size = 0.95 \[ \frac {2 b \tanh ^{-1}\left (\frac {\sqrt {-a-1} \sqrt {-a-b x+1}}{\sqrt {a-1} \sqrt {a+b x+1}}\right )}{(-a-1)^{3/2} \sqrt {a-1}}-\frac {\sqrt {-((a+b x-1) (a+b x+1))}}{(a+1) x} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 277, normalized size = 2.95 \[ \left [-\frac {\sqrt {-a^{2} + 1} b x \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{2 \, {\left (a^{3} + a^{2} - a - 1\right )} x}, -\frac {\sqrt {a^{2} - 1} b x \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{{\left (a^{3} + a^{2} - a - 1\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 223, normalized size = 2.37 \[ -\frac {2 \, b^{2} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{\sqrt {a^{2} - 1} {\left (a {\left | b \right |} + {\left | b \right |}\right )}} - \frac {2 \, {\left (a b^{2} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b}\right )}}{{\left (a^{2} {\left | b \right |} + a {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 565, normalized size = 6.01 \[ -\frac {b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (1+a \right )^{2}}+\frac {b^{2} a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\left (1+a \right )^{2} \sqrt {b^{2}}}+\frac {b \sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (1+a \right )^{2}}-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (1+a \right ) \left (-a^{2}+1\right ) x}-\frac {2 a b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (1+a \right ) \left (-a^{2}+1\right )}+\frac {a^{2} b^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\left (1+a \right ) \left (-a^{2}+1\right ) \sqrt {b^{2}}}+\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (1+a \right ) \sqrt {-a^{2}+1}}-\frac {b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x}{\left (1+a \right ) \left (-a^{2}+1\right )}-\frac {b^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\left (1+a \right ) \left (-a^{2}+1\right ) \sqrt {b^{2}}}+\frac {b \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{\left (1+a \right )^{2}}+\frac {b^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\left (1+a \right )^{2} \sqrt {b^{2}}} \]
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 \[ \int \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{{\left (b x + a + 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{x^2\,\left (a+b\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{x^{2} \left (a + b x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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