Optimal. Leaf size=93 \[ \frac {\sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \sqrt {1-a^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6164, 96, 93, 208} \[ \frac {\sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \sqrt {1-a^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 96
Rule 208
Rule 6164
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a+b x)}}{x \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx &=\int \frac {1}{x (1-a-b x)^{3/2} \sqrt {1+a+b x}} \, dx\\ &=\frac {\sqrt {1+a+b x}}{(1-a) \sqrt {1-a-b x}}+\frac {\int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{1-a}\\ &=\frac {\sqrt {1+a+b x}}{(1-a) \sqrt {1-a-b x}}+\frac {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 )}{1-a}\\ &=\frac {\sqrt {1+a+b x}}{(1-a) \sqrt {1-a-b x}}-\frac {2 \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 = 0.25, size = 118, normalized size = 1.27 \[ -\frac {-\frac {\sqrt {-a^2-2 a b x-b^2 x^2+1}}{a+b x-1}-\frac {\log \left (\sqrt {1-a^2} \sqrt {-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )}{\sqrt {1-a^2}}+\frac {\log (x)}{\sqrt {1-a^2}}}{a-1} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 316, normalized size = 3.40 \[ \left [-\frac {\sqrt {-a^{2} + 1} {\left (b x + a - 1\right )} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{2 \, {\left (a^{4} - 2 \, a^{3} + {\left (a^{3} - a^{2} - a + 1\right )} b x + 2 \, a - 1\right )}}, -\frac {\sqrt {a^{2} - 1} {\left (b x + a - 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{a^{4} - 2 \, a^{3} + {\left (a^{3} - a^{2} - a + 1\right )} b x + 2 \, a - 1}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 112, normalized size = 1.20 \[ -\frac {2 \, b \arctan \left (\frac {\frac {{\left (\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{\sqrt {a^{2} - 1} {\left (a {\left | b \right |} - {\left | b \right |}\right )}} - \frac {2 \, b}{{\left (a {\left | b \right |} - {\left | b \right |}\right )} {\left (\frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 391, normalized size = 4.20 \[ \frac {2 b \left (-2 b^{2} x -2 a b \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {1}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {a b x}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {a^{2}}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}+\frac {a}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {a^{2} b x}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {a^{3}}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a}{\left (-a^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {b x + a + 1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \sqrt {-{\left (b x + a\right )}^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {a+b\,x+1}{x\,\sqrt {1-{\left (a+b\,x\right )}^2}\,\left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________