Optimal. Leaf size=124 \[ \frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}}}{a^2}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{a^2 \sqrt {a x+1} \sqrt {1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.47, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6167, 6159, 6129, 80, 50, 41, 216} \[ \frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}}}{a^2}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{a^2 \sqrt {a x+1} \sqrt {1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 50
Rule 80
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-2 \tanh ^{-1}(a x)} x \sqrt {1-a x} \sqrt {1+a x} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {x (1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{3 a \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 84, normalized size = 0.68 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1} \left (a^2 x^2-3 a x+5\right )-3 \log \left (\sqrt {a^2 x^2-1}+a x\right )\right )}{3 a^2 \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 204, normalized size = 1.65 \[ \left [\frac {2 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{6 \, a^{3}}, \frac {{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{3 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 117, normalized size = 0.94 \[ \frac {1}{6} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (x {\left (\frac {x \mathrm {sgn}\relax (x)}{a^{2}} - \frac {3 \, \mathrm {sgn}\relax (x)}{a^{3}}\right )} + \frac {5 \, \mathrm {sgn}\relax (x)}{a^{4}}\right )} + \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a^{3} {\left | a \right |}} - \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 10 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\relax (x)}{a^{4} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 173, normalized size = 1.40 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{3}-3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2} c +3 c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-6 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right )+6 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a c \right )}{3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________