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 = 0.237261, antiderivative size = 78, normalized size of antiderivative = 1., 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 6152
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
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 = 0.126432, 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]
________________________________________________________________________________________
Maple [B] time = 0.042, size = 231, normalized size = 3. \begin{align*} 2\,{\frac{a \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}}{cx}}+2\,{a}^{3}x\sqrt{-{a}^{2}c{x}^{2}+c}+2\,{\frac{{a}^{3}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }-{\frac{3\,{a}^{2}}{2}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }+{\frac{3\,{a}^{2}}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}-2\,{a}^{2}\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }-2\,{\frac{{a}^{3}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}} \right ) }-{\frac{1}{2\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.35702, size = 339, normalized size = 4.35 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{4} + x^{3}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{4} + x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17133, size = 205, normalized size = 2.63 \begin{align*} \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}\left (a\right )}{\sqrt{-c}} - \frac{{\left (3 \, \pi a c - 8 \, a c\right )} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\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}\left (a\right ) - 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}\left (a\right )}{{\left (c - \frac{c}{a x + 1}\right )}^{2}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]