Optimal. Leaf size=98 \[ -\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a) \sqrt{1-a^2}}-\frac{\sqrt{-a-b x+1} \sqrt{a+b x+1}}{(1-a) x} \]
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Rubi [A] time = 0.0559057, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6163, 94, 93, 208} \[ -\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a) \sqrt{1-a^2}}-\frac{\sqrt{-a-b x+1} \sqrt{a+b x+1}}{(1-a) x} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 94
Rule 93
Rule 208
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*}
Mathematica [A] time = 0.0499374, size = 83, normalized size = 0.85 \[ \frac{\sqrt{-(a+b x-1) (a+b x+1)}}{(a-1) x}-\frac{2 b \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )}{(a-1)^{3/2} \sqrt{a+1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.063, size = 265, normalized size = 2.7 \begin{align*} -{\frac{1}{ \left ( -{a}^{2}+1 \right ) x}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{ab\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{a}{ \left ( -{a}^{2}+1 \right ) x}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{{a}^{2}b\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{b\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ){\frac{1}{\sqrt{-{a}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.66297, size = 633, normalized size = 6.46 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x + 1}{x^{2} \sqrt{- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27024, size = 257, normalized size = 2.62 \begin{align*} -\frac{2 \, b^{2} \arctan \left (\frac{\frac{{\left (\sqrt{-{\left (b x + a\right )}^{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{-{\left (b x + a\right )}^{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{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac{2 \,{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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