Optimal. Leaf size=85 \[ -\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6152, 1809, 780, 217, 203} \[ -\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 217
Rule 780
Rule 1809
Rule 6152
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx &=c \int \frac {x (1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x \left (-5 a^2 c+6 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{3 a^2}\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {c \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{a}\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {c \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{a}\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 80, normalized size = 0.94 \[ \frac {3 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )-\left (a^2 x^2-3 a x+5\right ) \sqrt {c-a^2 c x^2}}{3 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 150, normalized size = 1.76 \[ \left [-\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{6 \, a^{2}}, -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{3 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.41, size = 174, normalized size = 2.05 \[ \frac {{\left (24 \, a^{4} \sqrt {c} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - \frac {{\left (9 \, a^{4} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 3 \, a^{4} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 8 \, a^{4} c^{2} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)\right )} {\left (a x + 1\right )}^{3}}{c^{3}}\right )} {\left | a \right |}}{12 \, a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 154, normalized size = 1.81 \[ \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{2} c}+\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{a}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a \sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}{a^{2}}-\frac {2 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}\right )}{a \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 70, normalized size = 0.82 \[ \frac {\sqrt {-a^{2} c x^{2} + c} x}{a} - \frac {\sqrt {c} \arcsin \left (a x\right )}{a^{2}} - \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2}} + \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {x \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\right )\, dx - \int \frac {a x^{2} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________