Optimal. Leaf size=69 \[ \frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]
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Rubi [A] time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4619, 4677, 8} \[ \frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4619
Rule 4677
Rubi steps
\begin {align*} \int \sin ^{-1}(a x)^4 \, dx &=x \sin ^{-1}(a x)^4-(4 a) \int \frac {x \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4-12 \int \sin ^{-1}(a x)^2 \, dx\\ &=-12 x \sin ^{-1}(a x)^2+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4+(24 a) \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}-12 x \sin ^{-1}(a x)^2+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}-12 x \sin ^{-1}(a x)^2+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.00 \[ \frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac {24 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 55, normalized size = 0.80 \[ \frac {a x \arcsin \left (a x\right )^{4} - 12 \, a x \arcsin \left (a x\right )^{2} + 24 \, a x + 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 65, normalized size = 0.94 \[ x \arcsin \left (a x\right )^{4} - 12 \, x \arcsin \left (a x\right )^{2} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} + 24 \, x - \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 67, normalized size = 0.97 \[ \frac {a x \arcsin \left (a x \right )^{4}+4 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arcsin \left (a x \right )^{2}+24 a x -24 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 75, normalized size = 1.09 \[ x \arcsin \left (a x\right )^{4} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arcsin \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 48, normalized size = 0.70 \[ x\,\left ({\mathrm {asin}\left (a\,x\right )}^4-12\,{\mathrm {asin}\left (a\,x\right )}^2+24\right )+\frac {4\,\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}\,\left ({\mathrm {asin}\left (a\,x\right )}^2-6\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 65, normalized size = 0.94 \[ \begin {cases} x \operatorname {asin}^{4}{\left (a x \right )} - 12 x \operatorname {asin}^{2}{\left (a x \right )} + 24 x + \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{a} - \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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