Optimal. Leaf size=141 \[ \frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{24 a^3}-\frac {9 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {x}{3 a^2 \sin ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac {x^3}{2 \sin ^{-1}(a x)^2} \]
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Rubi [A] time = 0.30, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4633, 4719, 4631, 3299, 4621, 4723} \[ \frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{24 a^3}-\frac {9 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}-\frac {x}{3 a^2 \sin ^{-1}(a x)^2}+\frac {x^3}{2 \sin ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4621
Rule 4631
Rule 4633
Rule 4719
Rule 4723
Rubi steps
\begin {align*} \int \frac {x^2}{\sin ^{-1}(a x)^4} \, dx &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac {2 \int \frac {x}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx}{3 a}-a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {x}{3 a^2 \sin ^{-1}(a x)^2}+\frac {x^3}{2 \sin ^{-1}(a x)^2}-\frac {3}{2} \int \frac {x^2}{\sin ^{-1}(a x)^2} \, dx+\frac {\int \frac {1}{\sin ^{-1}(a x)^2} \, dx}{3 a^2}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {x}{3 a^2 \sin ^{-1}(a x)^2}+\frac {x^3}{2 \sin ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac {3 \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{4 x}+\frac {3 \sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{2 a^3}-\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)} \, dx}{3 a}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {x}{3 a^2 \sin ^{-1}(a x)^2}+\frac {x^3}{2 \sin ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {x}{3 a^2 \sin ^{-1}(a x)^2}+\frac {x^3}{2 \sin ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)}+\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{24 a^3}-\frac {9 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 102, normalized size = 0.72 \[ \frac {-\frac {8 a^2 x^2 \sqrt {1-a^2 x^2}}{\sin ^{-1}(a x)^3}+\frac {4 a x \left (3 a^2 x^2-2\right )}{\sin ^{-1}(a x)^2}+\frac {4 \sqrt {1-a^2 x^2} \left (9 a^2 x^2-2\right )}{\sin ^{-1}(a x)}+\text {Si}\left (\sin ^{-1}(a x)\right )-27 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{24 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arcsin \left (a x\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 148, normalized size = 1.05 \[ \frac {{\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )^{2}} - \frac {9 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{24 \, a^{3}} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )} + \frac {x}{6 \, a^{2} \arcsin \left (a x\right )^{2}} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{3} \arcsin \left (a x\right )} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 117, normalized size = 0.83 \[ \frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arcsin \left (a x \right )^{3}}+\frac {a x}{24 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )}+\frac {\Si \left (\arcsin \left (a x \right )\right )}{24}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {9 \Si \left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}} \]
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 \[ -\frac {a^{3} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3} \int \frac {{\left (27 \, a^{2} x^{3} - 20 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{{\left (a^{3} x^{2} - a\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} + {\left (2 \, a^{2} x^{2} - {\left (9 \, a^{2} x^{2} - 2\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - {\left (3 \, a^{3} x^{3} - 2 \, a x\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{6 \, a^{3} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}} \]
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 \[ \int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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