Optimal. Leaf size=134 \[ \frac {6 b \sqrt {1+a+b x}}{(1-a)^2 \sqrt {1-a-b x}}-\frac {(1+a+b x)^{3/2}}{(1-a) x \sqrt {1-a-b x}}-\frac {6 (1+a) b \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^2 \sqrt {1-a^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6298, 96, 95,
214} \begin {gather*} -\frac {6 (a+1) b \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a)^2 \sqrt {1-a^2}}-\frac {(a+b x+1)^{3/2}}{(1-a) x \sqrt {-a-b x+1}}+\frac {6 b \sqrt {a+b x+1}}{(1-a)^2 \sqrt {-a-b x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 214
Rule 6298
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1+a+b x)^{3/2}}{x^2 (1-a-b x)^{3/2}} \, dx\\ &=-\frac {(1+a+b x)^{3/2}}{(1-a) x \sqrt {1-a-b x}}+\frac {(3 b) \int \frac {\sqrt {1+a+b x}}{x (1-a-b x)^{3/2}} \, dx}{1-a}\\ &=\frac {6 b \sqrt {1+a+b x}}{(1-a)^2 \sqrt {1-a-b x}}-\frac {(1+a+b x)^{3/2}}{(1-a) x \sqrt {1-a-b x}}+\frac {(3 (1+a) b) \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{(1-a)^2}\\ &=\frac {6 b \sqrt {1+a+b x}}{(1-a)^2 \sqrt {1-a-b x}}-\frac {(1+a+b x)^{3/2}}{(1-a) x \sqrt {1-a-b x}}+\frac {(6 (1+a) b) \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)^2}\\ &=\frac {6 b \sqrt {1+a+b x}}{(1-a)^2 \sqrt {1-a-b x}}-\frac {(1+a+b x)^{3/2}}{(1-a) x \sqrt {1-a-b x}}-\frac {6 (1+a) b \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^2 \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 106, normalized size = 0.79 \begin {gather*} \frac {\sqrt {1+a+b x} \left (-1+a^2+5 b x+a b x\right )}{(-1+a)^2 x \sqrt {1-a-b x}}-\frac {6 \sqrt {-1-a} b \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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(688\) vs.
\(2(116)=232\).
time = 0.08, size = 689, normalized size = 5.14
method | result | size |
risch | \(\frac {\left (1+a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{\left (-1+a \right )^{2} x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}}{\left (-1+a \right )^{2} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {3 b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a}{\left (-1+a \right )^{2} \sqrt {-a^{2}+1}}-\frac {3 b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-1+a \right )^{2} \sqrt {-a^{2}+1}}\) | \(240\) |
default | \(b^{3} \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )+\frac {6 b^{2} a \left (-2 b^{2} x -2 b a \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {6 b^{2} \left (-2 b^{2} x -2 b a \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\left (a^{3}+3 a^{2}+3 a +1\right ) \left (-\frac {1}{\left (-a^{2}+1\right ) x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 b a \left (\frac {1}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {2 b a \left (-2 b^{2} x -2 b a \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{-a^{2}+1}+\frac {4 b^{2} \left (-2 b^{2} x -2 b a \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )+3 b \left (a^{2}+2 a +1\right ) \left (\frac {1}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {2 b a \left (-2 b^{2} x -2 b a \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(689\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 370, normalized size = 2.76 \begin {gather*} \left [\frac {3 \, {\left (b^{2} x^{2} + {\left (a - 1\right )} b x\right )} \sqrt {-\frac {a + 1}{a - 1}} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{3} + {\left (a^{2} - a\right )} b x - a^{2} - a + 1\right )} \sqrt {-\frac {a + 1}{a - 1}} + 2}{x^{2}}\right ) - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (a + 5\right )} b x + a^{2} - 1\right )}}{2 \, {\left ({\left (a^{2} - 2 \, a + 1\right )} b x^{2} + {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x\right )}}, \frac {3 \, {\left (b^{2} x^{2} + {\left (a - 1\right )} b x\right )} \sqrt {\frac {a + 1}{a - 1}} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {\frac {a + 1}{a - 1}}}{{\left (a + 1\right )} b^{2} x^{2} + a^{3} + 2 \, {\left (a^{2} + a\right )} b x + a^{2} - a - 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (a + 5\right )} b x + a^{2} - 1\right )}}{{\left (a^{2} - 2 \, a + 1\right )} b x^{2} + {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x}\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 {\left (a + b x + 1\right )^{3}}{x^{2} \left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, 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. 606 vs.
\(2 (109) = 218\).
time = 0.47, size = 606, normalized size = 4.52 \begin {gather*} \frac {6 \, {\left (a b^{2} + b^{2}\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^{2} {\left | b \right |} - 2 \, a {\left | b \right |} + {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {2 \, {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{2}}{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^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}} - 5 \, a^{2} b^{2} + \frac {10 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{2}}{b^{2} x + a b} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a b^{2}}{{\left (b^{2} x + a b\right )}^{2}} - 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} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}}{{\left (a^{3} {\left | b \right |} - 2 \, 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 )} a}{b^{2} x + a b} - \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}} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a}{{\left (b^{2} x + a b\right )}^{3}} - 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} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}} \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 {{\left (a+b\,x+1\right )}^3}{x^2\,{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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