Optimal. Leaf size=54 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0358068, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {93, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx &=2 \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 )\\ &=-\frac{2 \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 = 0.0480006, size = 88, normalized size = 1.63 \[ \frac{2 \sqrt{a+b x-1} \sqrt{a+b x+1} \tan ^{-1}\left (\frac{\sqrt{a+1} \sqrt{\frac{a+b x-1}{a+b x+1}}}{\sqrt{1-a}}\right )}{\sqrt{1-a^2} \sqrt{-(a+b x-1) (a+b x+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.04, size = 114, normalized size = 2.1 \begin{align*}{\frac{ \left ({\it csgn} \left ( b \right ) \right ) ^{2}}{ \left ( 1+a \right ) \left ( a-1 \right ) }\sqrt{-bx-a+1}\sqrt{bx+a+1}\sqrt{-{a}^{2}+1}\ln \left ( -2\,{\frac{abx+{a}^{2}-\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}-1}{x}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.22638, size = 440, normalized size = 8.15 \begin{align*} \left [-\frac{\sqrt{-a^{2} + 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 + 2 \,{\left (a b x + a^{2} - 1\right )} \sqrt{-a^{2} + 1} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} - 4 \, a^{2} + 2}{x^{2}}\right )}{2 \,{\left (a^{2} - 1\right )}}, \frac{\arctan \left (\frac{{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} \sqrt{b x + a + 1} \sqrt{-b x - a + 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{a^{2} - 1}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- a - b x + 1} \sqrt{a + b x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]