Optimal. Leaf size=370 \[ -\frac{6 c^3 \left (1-a^2 x^2\right )^{7/2}}{2401 a}-\frac{2664 c^3 \left (1-a^2 x^2\right )^{5/2}}{214375 a}-\frac{30256 c^3 \left (1-a^2 x^2\right )^{3/2}}{385875 a}-\frac{413312 c^3 \sqrt{1-a^2 x^2}}{128625 a}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3-\frac{4322 c^3 x \sin ^{-1}(a x)}{1225} \]
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Rubi [A] time = 0.699905, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {4649, 4619, 4677, 261, 4645, 444, 43, 194, 12, 1247, 698, 1799, 1850} \[ -\frac{6 c^3 \left (1-a^2 x^2\right )^{7/2}}{2401 a}-\frac{2664 c^3 \left (1-a^2 x^2\right )^{5/2}}{214375 a}-\frac{30256 c^3 \left (1-a^2 x^2\right )^{3/2}}{385875 a}-\frac{413312 c^3 \sqrt{1-a^2 x^2}}{128625 a}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3-\frac{4322 c^3 x \sin ^{-1}(a x)}{1225} \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4619
Rule 4677
Rule 261
Rule 4645
Rule 444
Rule 43
Rule 194
Rule 12
Rule 1247
Rule 698
Rule 1799
Rule 1850
Rubi steps
\begin{align*} \int \left (c-a^2 c x^2\right )^3 \sin ^{-1}(a x)^3 \, dx &=\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3+\frac{1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^3 \, dx-\frac{1}{7} \left (3 a c^3\right ) \int x \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2 \, dx\\ &=\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3+\frac{1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^3 \, dx-\frac{1}{49} \left (6 c^3\right ) \int \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x) \, dx-\frac{1}{35} \left (18 a c^3\right ) \int x \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2 \, dx\\ &=-\frac{6}{49} c^3 x \sin ^{-1}(a x)+\frac{6}{49} a^2 c^3 x^3 \sin ^{-1}(a x)-\frac{18}{245} a^4 c^3 x^5 \sin ^{-1}(a x)+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3-\frac{1}{175} \left (36 c^3\right ) \int \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x) \, dx+\frac{1}{35} \left (16 c^3\right ) \int \sin ^{-1}(a x)^3 \, dx+\frac{1}{49} \left (6 a c^3\right ) \int \frac{x \left (35-35 a^2 x^2+21 a^4 x^4-5 a^6 x^6\right )}{35 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{35} \left (24 a c^3\right ) \int x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \, dx\\ &=-\frac{402 c^3 x \sin ^{-1}(a x)}{1225}+\frac{318 a^2 c^3 x^3 \sin ^{-1}(a x)}{1225}-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3-\frac{1}{35} \left (16 c^3\right ) \int \left (1-a^2 x^2\right ) \sin ^{-1}(a x) \, dx+\frac{\left (6 a c^3\right ) \int \frac{x \left (35-35 a^2 x^2+21 a^4 x^4-5 a^6 x^6\right )}{\sqrt{1-a^2 x^2}} \, dx}{1715}+\frac{1}{175} \left (36 a c^3\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{35} \left (48 a c^3\right ) \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{962 c^3 x \sin ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3-\frac{1}{35} \left (96 c^3\right ) \int \sin ^{-1}(a x) \, dx+\frac{\left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{35-35 a^2 x+21 a^4 x^2-5 a^6 x^3}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )}{1715}+\frac{1}{875} \left (12 a c^3\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{1-a^2 x^2}} \, dx+\frac{1}{35} \left (16 a c^3\right ) \int \frac{x \left (1-\frac{a^2 x^2}{3}\right )}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4322 c^3 x \sin ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3+\frac{\left (3 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{16}{\sqrt{1-a^2 x}}+8 \sqrt{1-a^2 x}+6 \left (1-a^2 x\right )^{3/2}+5 \left (1-a^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{1715}+\frac{1}{875} \left (6 a c^3\right ) \operatorname{Subst}\left (\int \frac{15-10 a^2 x+3 a^4 x^2}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )+\frac{1}{35} \left (8 a c^3\right ) \operatorname{Subst}\left (\int \frac{1-\frac{a^2 x}{3}}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )+\frac{1}{35} \left (96 a c^3\right ) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{960 c^3 \sqrt{1-a^2 x^2}}{343 a}-\frac{16 c^3 \left (1-a^2 x^2\right )^{3/2}}{1715 a}-\frac{36 c^3 \left (1-a^2 x^2\right )^{5/2}}{8575 a}-\frac{6 c^3 \left (1-a^2 x^2\right )^{7/2}}{2401 a}-\frac{4322 c^3 x \sin ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3+\frac{1}{875} \left (6 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1-a^2 x}}+4 \sqrt{1-a^2 x}+3 \left (1-a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )+\frac{1}{35} \left (8 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-a^2 x}}+\frac{1}{3} \sqrt{1-a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{413312 c^3 \sqrt{1-a^2 x^2}}{128625 a}-\frac{30256 c^3 \left (1-a^2 x^2\right )^{3/2}}{385875 a}-\frac{2664 c^3 \left (1-a^2 x^2\right )^{5/2}}{214375 a}-\frac{6 c^3 \left (1-a^2 x^2\right )^{7/2}}{2401 a}-\frac{4322 c^3 x \sin ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{18 c^3 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.322504, size = 171, normalized size = 0.46 \[ \frac{c^3 \left (2 \sqrt{1-a^2 x^2} \left (16875 a^6 x^6-134541 a^4 x^4+747937 a^2 x^2-22329151\right )-385875 a x \left (5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right ) \sin ^{-1}(a x)^3-11025 \sqrt{1-a^2 x^2} \left (75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right ) \sin ^{-1}(a x)^2+210 a x \left (1125 a^6 x^6-7371 a^4 x^4+26495 a^2 x^2-226905\right ) \sin ^{-1}(a x)\right )}{13505625 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 278, normalized size = 0.8 \begin{align*} -{\frac{{c}^{3}}{13505625\,a} \left ( 1929375\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}{a}^{7}{x}^{7}+826875\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}{a}^{6}{x}^{6}-8103375\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}{a}^{5}{x}^{5}-236250\,\arcsin \left ( ax \right ){a}^{7}{x}^{7}-3869775\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}{a}^{4}{x}^{4}-33750\,{a}^{6}{x}^{6}\sqrt{-{a}^{2}{x}^{2}+1}+13505625\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}+1547910\,{a}^{5}{x}^{5}\arcsin \left ( ax \right ) +8345925\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}{a}^{2}{x}^{2}+269082\,{a}^{4}{x}^{4}\sqrt{-{a}^{2}{x}^{2}+1}-13505625\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}-5563950\,{a}^{3}{x}^{3}\arcsin \left ( ax \right ) -23825025\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-1495874\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}+47650050\,ax\arcsin \left ( ax \right ) +44658302\,\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56688, size = 383, normalized size = 1.04 \begin{align*} -\frac{1}{1225} \,{\left (75 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{3} x^{6} - 351 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 757 \, \sqrt{-a^{2} x^{2} + 1} c^{3} x^{2} - \frac{2161 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} - \frac{1}{35} \,{\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \arcsin \left (a x\right )^{3} + \frac{2}{13505625} \,{\left (16875 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{3} x^{6} - 134541 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt{-a^{2} x^{2} + 1} c^{3} x^{2} - \frac{22329151 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{a^{2}} + \frac{105 \,{\left (1125 \, a^{6} c^{3} x^{7} - 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} - 226905 \, c^{3} x\right )} \arcsin \left (a x\right )}{a}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79716, size = 512, normalized size = 1.38 \begin{align*} -\frac{385875 \,{\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \arcsin \left (a x\right )^{3} - 210 \,{\left (1125 \, a^{7} c^{3} x^{7} - 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} - 226905 \, a c^{3} x\right )} \arcsin \left (a x\right ) -{\left (33750 \, a^{6} c^{3} x^{6} - 269082 \, a^{4} c^{3} x^{4} + 1495874 \, a^{2} c^{3} x^{2} - 44658302 \, c^{3} - 11025 \,{\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \arcsin \left (a x\right )^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{13505625 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.7385, size = 355, normalized size = 0.96 \begin{align*} \begin{cases} - \frac{a^{6} c^{3} x^{7} \operatorname{asin}^{3}{\left (a x \right )}}{7} + \frac{6 a^{6} c^{3} x^{7} \operatorname{asin}{\left (a x \right )}}{343} - \frac{3 a^{5} c^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{49} + \frac{6 a^{5} c^{3} x^{6} \sqrt{- a^{2} x^{2} + 1}}{2401} + \frac{3 a^{4} c^{3} x^{5} \operatorname{asin}^{3}{\left (a x \right )}}{5} - \frac{702 a^{4} c^{3} x^{5} \operatorname{asin}{\left (a x \right )}}{6125} + \frac{351 a^{3} c^{3} x^{4} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{1225} - \frac{29898 a^{3} c^{3} x^{4} \sqrt{- a^{2} x^{2} + 1}}{1500625} - a^{2} c^{3} x^{3} \operatorname{asin}^{3}{\left (a x \right )} + \frac{1514 a^{2} c^{3} x^{3} \operatorname{asin}{\left (a x \right )}}{3675} - \frac{757 a c^{3} x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{1225} + \frac{1495874 a c^{3} x^{2} \sqrt{- a^{2} x^{2} + 1}}{13505625} + c^{3} x \operatorname{asin}^{3}{\left (a x \right )} - \frac{4322 c^{3} x \operatorname{asin}{\left (a x \right )}}{1225} + \frac{2161 c^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{1225 a} - \frac{44658302 c^{3} \sqrt{- a^{2} x^{2} + 1}}{13505625 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45968, size = 512, normalized size = 1.38 \begin{align*} -\frac{1}{7} \,{\left (a^{2} x^{2} - 1\right )}^{3} c^{3} x \arcsin \left (a x\right )^{3} + \frac{6}{35} \,{\left (a^{2} x^{2} - 1\right )}^{2} c^{3} x \arcsin \left (a x\right )^{3} + \frac{6}{343} \,{\left (a^{2} x^{2} - 1\right )}^{3} c^{3} x \arcsin \left (a x\right ) - \frac{8}{35} \,{\left (a^{2} x^{2} - 1\right )} c^{3} x \arcsin \left (a x\right )^{3} - \frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt{-a^{2} x^{2} + 1} c^{3} \arcsin \left (a x\right )^{2}}{49 \, a} - \frac{2664}{42875} \,{\left (a^{2} x^{2} - 1\right )}^{2} c^{3} x \arcsin \left (a x\right ) + \frac{16}{35} \, c^{3} x \arcsin \left (a x\right )^{3} + \frac{18 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} c^{3} \arcsin \left (a x\right )^{2}}{175 \, a} + \frac{30256}{128625} \,{\left (a^{2} x^{2} - 1\right )} c^{3} x \arcsin \left (a x\right ) + \frac{6 \,{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt{-a^{2} x^{2} + 1} c^{3}}{2401 \, a} + \frac{8 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3} \arcsin \left (a x\right )^{2}}{35 \, a} - \frac{413312}{128625} \, c^{3} x \arcsin \left (a x\right ) - \frac{2664 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} c^{3}}{214375 \, a} + \frac{48 \, \sqrt{-a^{2} x^{2} + 1} c^{3} \arcsin \left (a x\right )^{2}}{35 \, a} - \frac{30256 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3}}{385875 \, a} - \frac{413312 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{128625 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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