Optimal. Leaf size=198 \[ -\frac {3 \sin ^{-1}(a x)^4}{32 a^4}+\frac {45 \sin ^{-1}(a x)^2}{128 a^4}+\frac {45 x^2}{128 a^2}-\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^2}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^3}-\frac {45 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{64 a^3}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4-\frac {3}{16} x^4 \sin ^{-1}(a x)^2+\frac {3 x^4}{128} \]
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Rubi [A] time = 0.52, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4627, 4707, 4641, 30} \[ \frac {45 x^2}{128 a^2}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a}-\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^2}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^3}-\frac {45 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{64 a^3}-\frac {3 \sin ^{-1}(a x)^4}{32 a^4}+\frac {45 \sin ^{-1}(a x)^2}{128 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4-\frac {3}{16} x^4 \sin ^{-1}(a x)^2+\frac {3 x^4}{128} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4627
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int x^3 \sin ^{-1}(a x)^4 \, dx &=\frac {1}{4} x^4 \sin ^{-1}(a x)^4-a \int \frac {x^4 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4-\frac {3}{4} \int x^3 \sin ^{-1}(a x)^2 \, dx-\frac {3 \int \frac {x^2 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {3}{16} x^4 \sin ^{-1}(a x)^2+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4-\frac {3 \int \frac {\sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}-\frac {9 \int x \sin ^{-1}(a x)^2 \, dx}{8 a^2}+\frac {1}{8} (3 a) \int \frac {x^4 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a}-\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^2}-\frac {3}{16} x^4 \sin ^{-1}(a x)^2+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}-\frac {3 \sin ^{-1}(a x)^4}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4+\frac {3 \int x^3 \, dx}{32}+\frac {9 \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{32 a}+\frac {9 \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=\frac {3 x^4}{128}-\frac {45 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a}-\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^2}-\frac {3}{16} x^4 \sin ^{-1}(a x)^2+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}-\frac {3 \sin ^{-1}(a x)^4}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4+\frac {9 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 a^3}+\frac {9 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}+\frac {9 \int x \, dx}{64 a^2}+\frac {9 \int x \, dx}{16 a^2}\\ &=\frac {45 x^2}{128 a^2}+\frac {3 x^4}{128}-\frac {45 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{32 a}+\frac {45 \sin ^{-1}(a x)^2}{128 a^4}-\frac {9 x^2 \sin ^{-1}(a x)^2}{16 a^2}-\frac {3}{16} x^4 \sin ^{-1}(a x)^2+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a}-\frac {3 \sin ^{-1}(a x)^4}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A] time = 0.06, size = 135, normalized size = 0.68 \[ \frac {4 \left (8 a^4 x^4-3\right ) \sin ^{-1}(a x)^4+3 a^2 x^2 \left (a^2 x^2+15\right )+16 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)^3-6 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+15\right ) \sin ^{-1}(a x)-3 \left (8 a^4 x^4+24 a^2 x^2-15\right ) \sin ^{-1}(a x)^2}{128 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 121, normalized size = 0.61 \[ \frac {3 \, a^{4} x^{4} + 4 \, {\left (8 \, a^{4} x^{4} - 3\right )} \arcsin \left (a x\right )^{4} + 45 \, a^{2} x^{2} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right )^{2} + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )\right )}}{128 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 234, normalized size = 1.18 \[ -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{3}}{4 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{4}}{4 \, a^{4}} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{8 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{4}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{32 \, a^{3}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{16 \, a^{4}} + \frac {5 \, \arcsin \left (a x\right )^{4}}{32 \, a^{4}} - \frac {51 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{64 \, a^{3}} - \frac {15 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{4}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{128 \, a^{4}} - \frac {51 \, \arcsin \left (a x\right )^{2}}{128 \, a^{4}} + \frac {51 \, {\left (a^{2} x^{2} - 1\right )}}{128 \, a^{4}} + \frac {195}{1024 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 209, normalized size = 1.06 \[ \frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{4}}{4}-\frac {\arcsin \left (a x \right )^{3} \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{8}-\frac {3 a^{4} x^{4} \arcsin \left (a x \right )^{2}}{16}+\frac {3 \arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{64}+\frac {27 \arcsin \left (a x \right )^{2}}{128}+\frac {3 a^{4} x^{4}}{128}+\frac {45 a^{2} x^{2}}{128}-\frac {9 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )^{2}}{16}-\frac {9 \arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{16}+\frac {9 \arcsin \left (a x \right )^{4}}{32}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{4} + a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{4} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\mathrm {asin}\left (a\,x\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.10, size = 190, normalized size = 0.96 \[ \begin {cases} \frac {x^{4} \operatorname {asin}^{4}{\left (a x \right )}}{4} - \frac {3 x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{16} + \frac {3 x^{4}}{128} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{4 a} - \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{32 a} - \frac {9 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{2}} + \frac {45 x^{2}}{128 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{8 a^{3}} - \frac {45 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{64 a^{3}} - \frac {3 \operatorname {asin}^{4}{\left (a x \right )}}{32 a^{4}} + \frac {45 \operatorname {asin}^{2}{\left (a x \right )}}{128 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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