Optimal. Leaf size=78 \[ \frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6152, 1807, 807, 266, 63, 208} \[ \frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6152
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx &=c \int \frac {(1-a x)^2}{x^3 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {1}{2} \int \frac {4 a c-3 a^2 c x}{x^2 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {1}{2} \left (3 a^2 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {1}{4} \left (3 a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 76, normalized size = 0.97 \[ \frac {1}{2} \left (\frac {(4 a x-1) \sqrt {c-a^2 c x^2}}{x^2}-3 a^2 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+3 a^2 \sqrt {c} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 149, normalized size = 1.91 \[ \left [\frac {3 \, a^{2} \sqrt {c} x^{2} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 1\right )}}{4 \, x^{2}}, -\frac {3 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 1\right )}}{2 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 3.09, size = 152, normalized size = 1.95 \[ \frac {1}{4} \, {\left (\frac {12 \, a c \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c}} - \frac {{\left (3 \, \pi a c - 8 \, a c\right )} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c}} + \frac {3 \, a c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 5 \, a c {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{{\left (c - \frac {c}{a x + 1}\right )}^{2}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 231, normalized size = 2.96 \[ -\frac {3 \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right ) a^{2}}{2}+\frac {3 \sqrt {-a^{2} c \,x^{2}+c}\, a^{2}}{2}+\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}+2 a^{3} x \sqrt {-a^{2} c \,x^{2}+c}+\frac {2 a^{3} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-2 a^{2} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}-\frac {2 a^{3} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{x^3\,{\left (a\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{4} + x^{3}}\right )\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{4} + x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________