Optimal. Leaf size=61 \[ -\frac{x^7}{49}+\frac{8 x^5}{175}-\frac{x^3}{105}+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac{2 x}{35} \]
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Rubi [A] time = 0.0750953, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {266, 43, 4690, 12, 373} \[ -\frac{x^7}{49}+\frac{8 x^5}{175}-\frac{x^3}{105}+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac{2 x}{35} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 4690
Rule 12
Rule 373
Rubi steps
\begin{align*} \int x^3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x) \, dx &=-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)+\int \frac{1}{35} \left (-2-5 x^2\right ) \left (1-x^2\right )^2 \, dx\\ &=-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)+\frac{1}{35} \int \left (-2-5 x^2\right ) \left (1-x^2\right )^2 \, dx\\ &=-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)+\frac{1}{35} \int \left (-2-x^2+8 x^4-5 x^6\right ) \, dx\\ &=-\frac{2 x}{35}-\frac{x^3}{105}+\frac{8 x^5}{175}-\frac{x^7}{49}-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.057314, size = 47, normalized size = 0.77 \[ -\frac{x \left (75 x^6-168 x^4+35 x^2+210\right )}{3675}-\frac{1}{35} \left (5 x^2+2\right ) \left (1-x^2\right )^{5/2} \cos ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.28, size = 430, normalized size = 7.1 \begin{align*}{\frac{i+7\,\arccos \left ( x \right ) }{6272} \left ( 64\,i{x}^{7}-64\,\sqrt{-{x}^{2}+1}{x}^{6}-112\,i{x}^{5}+80\,\sqrt{-{x}^{2}+1}{x}^{4}+56\,i{x}^{3}-24\,\sqrt{-{x}^{2}+1}{x}^{2}-7\,ix+\sqrt{-{x}^{2}+1} \right ) }-{\frac{i+5\,\arccos \left ( x \right ) }{3200} \left ( 16\,i{x}^{5}-16\,\sqrt{-{x}^{2}+1}{x}^{4}-20\,i{x}^{3}+12\,\sqrt{-{x}^{2}+1}{x}^{2}+5\,ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{i+3\,\arccos \left ( x \right ) }{384} \left ( 4\,i{x}^{3}-4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{3\,\arccos \left ( x \right ) +3\,i}{128} \left ( ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{3\,\arccos \left ( x \right ) -3\,i}{128} \left ( ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+3\,\arccos \left ( x \right ) }{384} \left ( 4\,i{x}^{3}+4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix-\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+5\,\arccos \left ( x \right ) }{3200} \left ( 16\,i{x}^{5}+16\,\sqrt{-{x}^{2}+1}{x}^{4}-20\,i{x}^{3}-12\,\sqrt{-{x}^{2}+1}{x}^{2}+5\,ix+\sqrt{-{x}^{2}+1} \right ) }-{\frac{-i+7\,\arccos \left ( x \right ) }{6272} \left ( 64\,i{x}^{7}+64\,\sqrt{-{x}^{2}+1}{x}^{6}-112\,i{x}^{5}-80\,\sqrt{-{x}^{2}+1}{x}^{4}+56\,i{x}^{3}+24\,\sqrt{-{x}^{2}+1}{x}^{2}-7\,ix-\sqrt{-{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40885, size = 66, normalized size = 1.08 \begin{align*} -\frac{1}{49} \, x^{7} + \frac{8}{175} \, x^{5} - \frac{1}{105} \, x^{3} - \frac{1}{35} \,{\left (5 \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x^{2} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}}\right )} \arccos \left (x\right ) - \frac{2}{35} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45917, size = 138, normalized size = 2.26 \begin{align*} -\frac{1}{49} \, x^{7} + \frac{8}{175} \, x^{5} - \frac{1}{105} \, x^{3} - \frac{1}{35} \,{\left (5 \, x^{6} - 8 \, x^{4} + x^{2} + 2\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - \frac{2}{35} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 153.986, size = 88, normalized size = 1.44 \begin{align*} - \frac{x^{7}}{49} - \frac{x^{6} \sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{7} + \frac{8 x^{5}}{175} + \frac{8 x^{4} \sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{35} - \frac{x^{3}}{105} - \frac{x^{2} \sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{35} - \frac{2 x}{35} - \frac{2 \sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{35} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07375, size = 81, normalized size = 1.33 \begin{align*} -\frac{1}{49} \, x^{7} + \frac{8}{175} \, x^{5} - \frac{1}{105} \, x^{3} - \frac{1}{35} \,{\left (5 \,{\left (x^{2} - 1\right )}^{3} \sqrt{-x^{2} + 1} + 7 \,{\left (x^{2} - 1\right )}^{2} \sqrt{-x^{2} + 1}\right )} \arccos \left (x\right ) - \frac{2}{35} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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