Optimal. Leaf size=162 \[ \frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^2 \sqrt {1-a^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6298, 98, 96,
95, 214} \begin {gather*} -\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) (a+1)^2 \sqrt {1-a^2}}-\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}+\frac {(1-2 a) b \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 (1-a) (a+1)^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 214
Rule 6298
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {\sqrt {1-a-b x}}{x^3 \sqrt {1+a+b x}} \, dx\\ &=-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {((1-2 a) b) \int \frac {\sqrt {1-a-b x}}{x^2 \sqrt {1+a+b x}} \, dx}{2 \left (1-a^2\right )}\\ &=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}+\frac {\left ((1-2 a) b^2\right ) \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 (1-a) (1+a)^2}\\ &=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}+\frac {\left ((1-2 a) b^2\right ) \text {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) (1+a)^2}\\ &=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^2 \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 122, normalized size = 0.75 \begin {gather*} -\frac {\left (-1+a^2+2 b x-a b x\right ) \sqrt {1-a^2-2 a b x-b^2 x^2}}{2 (-1+a) (1+a)^2 x^2}+\frac {(-1+2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{5/2} (-1+a)^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1045\) vs.
\(2(140)=280\).
time = 0.07, size = 1046, normalized size = 6.46
method | result | size |
risch | \(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (-a b x +a^{2}+2 b x -1\right )}{2 \left (1+a \right ) x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a}{\left (a^{2}-1\right ) \left (1+a \right ) \sqrt {-a^{2}+1}}+\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (a^{2}-1\right ) \left (1+a \right ) \sqrt {-a^{2}+1}}\) | \(225\) |
default | \(-\frac {b \left (-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (-a^{2}+1\right ) x}-\frac {b a \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b 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 )}{\sqrt {b^{2}}}-\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 )\right )}{-a^{2}+1}-\frac {2 b^{2} \left (-\frac {\left (-2 b^{2} x -2 b a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \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 )}{8 b^{2} \sqrt {b^{2}}}\right )}{-a^{2}+1}\right )}{\left (1+a \right )^{2}}-\frac {b^{2} \left (\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}+\frac {b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\sqrt {b^{2}}}\right )}{\left (1+a \right )^{3}}+\frac {b^{2} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b 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 )}{\sqrt {b^{2}}}-\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 )\right )}{\left (1+a \right )^{3}}+\frac {-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {b a \left (-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (-a^{2}+1\right ) x}-\frac {b a \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b 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 )}{\sqrt {b^{2}}}-\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 )\right )}{-a^{2}+1}-\frac {2 b^{2} \left (-\frac {\left (-2 b^{2} x -2 b a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \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 )}{8 b^{2} \sqrt {b^{2}}}\right )}{-a^{2}+1}\right )}{-2 a^{2}+2}-\frac {b^{2} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b 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 )}{\sqrt {b^{2}}}-\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 )\right )}{2 \left (-a^{2}+1\right )}}{1+a}\) | \(1046\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 356, normalized size = 2.20 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \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 \, {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \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 ) - {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{x^{3} \left (a + b x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 750 vs.
\(2 (134) = 268\).
time = 0.42, size = 750, normalized size = 4.63 \begin {gather*} \frac {{\left (2 \, a b^{3} - b^{3}\right )} \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 )}{{\left (a^{3} {\left | b \right |} + a^{2} {\left | b \right |} - a {\left | b \right |} - {\left | b \right |}\right )} \sqrt {a^{2} - 1}} + \frac {\frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{4} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + 2 \, a^{4} b^{3} - \frac {5 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{3} b^{3}}{b^{2} x + a b} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} - \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - 2 \, a^{3} b^{3} + \frac {6 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{3}}{b^{2} x + a b} + \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - a^{2} b^{3} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{3}}{b^{2} x + a b} - \frac {4 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}}}{{\left (a^{5} {\left | b \right |} + a^{4} {\left | b \right |} - a^{3} {\left | b \right |} - a^{2} {\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 )}^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{x^3\,\left (a+b\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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