Optimal. Leaf size=64 \[ \sin ^{-1}(a+b x)-\frac{2 (a+1) \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]
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Rubi [A] time = 0.0754015, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6163, 105, 53, 619, 216, 93, 208} \[ \sin ^{-1}(a+b x)-\frac{2 (a+1) \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 105
Rule 53
Rule 619
Rule 216
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a+b x)}}{x} \, dx &=\int \frac{\sqrt{1+a+b x}}{x \sqrt{1-a-b x}} \, dx\\ &=-\left ((-1-a) \int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx\right )+b \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx\\ &=(2 (1+a)) \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 )+b \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx\\ &=-\frac{2 (1+a) \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{\sqrt{1-a^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{2 b}\\ &=\sin ^{-1}(a+b x)-\frac{2 (1+a) \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{\sqrt{1-a^2}}\\ \end{align*}
Mathematica [A] time = 0.10338, size = 100, normalized size = 1.56 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )}{\sqrt{\frac{a-1}{a+1}}}+\frac{2 \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.057, size = 168, normalized size = 2.6 \begin{align*}{b\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\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}}}}-{a\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 [B] time = 2.63945, size = 717, normalized size = 11.2 \begin{align*} \left [\frac{1}{2} \, \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 ) - \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\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 ) - \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )\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 \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 [A] time = 1.20983, size = 107, normalized size = 1.67 \begin{align*} -\frac{b \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} + \frac{2 \,{\left (a b + b\right )} \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 | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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