Optimal. Leaf size=112 \[ \frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.396259, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6167, 6152, 1809, 833, 780, 217, 203} \[ \frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6152
Rule 1809
Rule 833
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac{x^2 (1-a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{\int \frac{x^2 \left (-7 a^2 c+8 a^3 c x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{4 a^2}\\ &=-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{\int \frac{x \left (-16 a^3 c^2+21 a^4 c^2 x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{12 a^4 c}\\ &=-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{(7 c) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{8 a^2}\\ &=-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{(7 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 )}{8 a^2}\\ &=-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.0994461, size = 88, normalized size = 0.79 \[ \frac{\left (6 a^3 x^3-16 a^2 x^2+21 a x-32\right ) \sqrt{c-a^2 c x^2}+21 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{24 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 178, normalized size = 1.6 \begin{align*} -{\frac{x}{4\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{8\,{a}^{2}}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{9\,c}{8\,{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2}{3\,{a}^{3}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}{{a}^{3}}}-2\,{\frac{c}{{a}^{2}\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62629, size = 396, normalized size = 3.54 \begin{align*} \left [\frac{2 \,{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt{-a^{2} c x^{2} + c} + 21 \, \sqrt{-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right )}{48 \, a^{3}}, \frac{{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt{-a^{2} c x^{2} + c} + 21 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\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{x^{2} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13788, size = 113, normalized size = 1.01 \begin{align*} \frac{1}{24} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (3 \, x - \frac{8}{a}\right )} x + \frac{21}{a^{2}}\right )} x - \frac{32}{a^{3}}\right )} + \frac{7 \, c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{2} \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]