Optimal. Leaf size=99 \[ -\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}-\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}-\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt {1-a x} \sqrt {a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6167, 6159, 6129, 50, 41, 216} \[ -\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}-\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}-\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 50
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 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-2 \tanh ^{-1}(a 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 {(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 a}-\frac {\left (3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}-\frac {\left (3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}-\frac {\left (3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{2 a \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 100, normalized size = 1.01 \[ -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a x+1} \left (a^2 x^2-5 a x+4\right )-6 \sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 188, normalized size = 1.90 \[ \left [\frac {2 \, {\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{4 \, a^{2}}, \frac {{\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{2 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 106, normalized size = 1.07 \[ \frac {1}{4} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (\frac {x \mathrm {sgn}\relax (x)}{a^{2}} - \frac {4 \, \mathrm {sgn}\relax (x)}{a^{3}}\right )} - \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a^{2} {\left | a \right |}} + \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 8 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\relax (x)}{a^{3} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 147, normalized size = 1.48 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-\sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )+4 \sqrt {c}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right )-4 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a \right )}{2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________