Optimal. Leaf size=130 \[ \frac{6 (1-a) b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(a+1)^2 \sqrt{1-a^2}}-\frac{(-a-b x+1)^{3/2}}{(a+1) x \sqrt{a+b x+1}}-\frac{6 b \sqrt{-a-b x+1}}{(a+1)^2 \sqrt{a+b x+1}} \]
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Rubi [A] time = 0.0768065, antiderivative size = 130, normalized size of antiderivative = 1., 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 = {6163, 94, 93, 208} \[ \frac{6 (1-a) b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(a+1)^2 \sqrt{1-a^2}}-\frac{(-a-b x+1)^{3/2}}{(a+1) x \sqrt{a+b x+1}}-\frac{6 b \sqrt{-a-b x+1}}{(a+1)^2 \sqrt{a+b x+1}} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 94
Rule 93
Rule 208
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) \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)^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*}
Mathematica [A] time = 0.0984897, size = 101, normalized size = 0.78 \[ \frac{\sqrt{-a-b x+1} \left (a^2+a b x-5 b x-1\right )}{(a+1)^2 x \sqrt{a+b x+1}}+\frac{6 \sqrt{a-1} b \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )}{(a+1)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.08, size = 1710, normalized size = 13.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b x + a + 1\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82437, size = 871, normalized size = 6.7 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22405, size = 815, normalized size = 6.27 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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