Optimal. Leaf size=84 \[ -\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}-\frac {(2 a+1) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^2}-\frac {(2 a+1) \sin ^{-1}(a+b x)}{2 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 80, 50, 53, 619, 216} \[ -\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}-\frac {(2 a+1) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^2}-\frac {(2 a+1) \sin ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 53
Rule 80
Rule 216
Rule 619
Rule 6163
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a+b x)} x \, dx &=\int \frac {x \sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx\\ &=-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}-\frac {(1+2 a) \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx}{2 b}\\ &=-\frac {(1+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}-\frac {(1+2 a) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b}\\ &=-\frac {(1+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}-\frac {(1+2 a) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b}\\ &=-\frac {(1+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}+\frac {(1+2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^3}\\ &=-\frac {(1+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}-\frac {(1+2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 99, normalized size = 1.18 \[ \frac {\sqrt {a+b x+1} \left (a^2+a-b^2 x^2+3 b x-2\right )}{2 b^2 \sqrt {-a-b x+1}}+\frac {(2 a+1) \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {b}}\right )}{(-b)^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 91, normalized size = 1.08 \[ \frac {{\left (2 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - a - 2\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 68, normalized size = 0.81 \[ \frac {1}{2} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\frac {x}{b} - \frac {a b + 2 \, b}{b^{3}}\right )} + \frac {{\left (2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 302, normalized size = 3.60 \[ \frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b}+\frac {a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b \sqrt {b^{2}}}-\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b^{2}}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{b \sqrt {b^{2}}}-\frac {a \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{b \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 107, normalized size = 1.27 \[ \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{2 \, b} - \frac {a \arcsin \left (b x + a\right )}{b^{2}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{2 \, b^{2}} - \frac {\arcsin \left (b x + a\right )}{2 \, b^{2}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sqrt {1-{\left (a+b\,x\right )}^2}}{a+b\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{a + b x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________