Optimal. Leaf size=109 \[ \frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} \sqrt {1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac {3 \left (1-2 a+2 a^2\right ) \text {ArcSin}(a+b x)}{2 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6299, 100, 152,
55, 633, 222} \begin {gather*} -\frac {3 \left (2 a^2-2 a+1\right ) \text {ArcSin}(a+b x)}{2 b^4}+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^4}+\frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 55
Rule 100
Rule 152
Rule 222
Rule 633
Rule 6299
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx &=\int \frac {x^3}{(1-a-b x)^{3/2} \sqrt {1+a+b x}} \, dx\\ &=\frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}-\frac {\int \frac {x \left (2 \left (1-a^2\right )+(3-2 a) b x\right )}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{b^2}\\ &=\frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} \sqrt {1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac {\left (3 \left (1-2 a+2 a^2\right )\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b^3}\\ &=\frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} \sqrt {1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac {\left (3 \left (1-2 a+2 a^2\right )\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^3}\\ &=\frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} \sqrt {1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}+\frac {\left (3 \left (1-2 a+2 a^2\right )\right ) \text {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^5}\\ &=\frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} \sqrt {1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac {3 \left (1-2 a+2 a^2\right ) \sin ^{-1}(a+b x)}{2 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 90, normalized size = 0.83 \begin {gather*} -\frac {-\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (-4-11 a^2+2 a^3+b x+b^2 x^2+a (13-4 b x)\right )}{-1+a+b x}+3 \left (1-2 a+2 a^2\right ) \text {ArcSin}(a+b x)}{2 b^4} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(839\) vs.
\(2(97)=194\).
time = 0.08, size = 840, normalized size = 7.71
method | result | size |
risch | \(\frac {\left (-b x +5 a -2\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{2 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}}{b^{3} \sqrt {b^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a}{b^{3} \sqrt {b^{2}}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{3} \sqrt {b^{2}}}+\frac {\sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a^{3}}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {3 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a^{2}}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}+\frac {3 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {\sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}\) | \(403\) |
default | \(b \left (-\frac {x^{3}}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {5 a \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2}}\right )}{2 b}+\frac {3 \left (-a^{2}+1\right ) \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )}{2 b^{2}}\right )+\left (1+a \right ) \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2}}\right )\) | \(840\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 576 vs.
\(2 (95) = 190\).
time = 0.58, size = 576, normalized size = 5.28 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{b^{6} x + a b^{5} - b^{5}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} b}{b^{7} x + a b^{6} + b^{6}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} b}{b^{7} x + a b^{6} - b^{6}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{6} x + a b^{5} + b^{5}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{6} x + a b^{5} - b^{5}} + \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{6} x + a b^{5} - b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b}{b^{7} x + a b^{6} + b^{6}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b}{b^{7} x + a b^{6} - b^{6}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{6} x + a b^{5} + b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{6} x + a b^{5} - b^{5}} - \frac {6 \, a^{2} \arcsin \left (b x + a\right )}{b^{5}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{4}} + \frac {6 \, a \arcsin \left (b x + a\right )}{b^{5}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{5}} - \frac {3 \, \arcsin \left (b x + a\right )}{b^{5}} + \frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{5}}\right )} b^{2}}{2 \, \sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 150, normalized size = 1.38 \begin {gather*} \frac {3 \, {\left (2 \, a^{3} + {\left (2 \, a^{2} - 2 \, a + 1\right )} b x - 4 \, a^{2} + 3 \, 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 ) + {\left (b^{2} x^{2} + 2 \, a^{3} - {\left (4 \, a - 1\right )} b x - 11 \, a^{2} + 13 \, a - 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (b^{5} x + {\left (a - 1\right )} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{3}}{a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 143, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\frac {x}{b^{3}} - \frac {5 \, a b^{6} - 2 \, b^{6}}{b^{10}}\right )} + \frac {3 \, {\left (2 \, a^{2} - 2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b^{3} {\left | b \right |}} - \frac {2 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )}}{b^{3} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^3\,\left (a+b\,x+1\right )}{\sqrt {1-{\left (a+b\,x\right )}^2}\,\left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________