Optimal. Leaf size=97 \[ -\frac {2 \text {Ci}\left (2 \sin ^{-1}(a x)\right )}{3 a^2}+\frac {2 x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)}-\frac {x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {1}{6 a^2 \sin ^{-1}(a x)^2}+\frac {x^2}{3 \sin ^{-1}(a x)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4633, 4719, 4631, 3302, 4641} \[ -\frac {2 \text {CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{3 a^2}+\frac {2 x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)}-\frac {x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {1}{6 a^2 \sin ^{-1}(a x)^2}+\frac {x^2}{3 \sin ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4631
Rule 4633
Rule 4641
Rule 4719
Rubi steps
\begin {align*} \int \frac {x}{\sin ^{-1}(a x)^4} \, dx &=-\frac {x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx}{3 a}-\frac {1}{3} (2 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {1}{6 a^2 \sin ^{-1}(a x)^2}+\frac {x^2}{3 \sin ^{-1}(a x)^2}-\frac {2}{3} \int \frac {x}{\sin ^{-1}(a x)^2} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {1}{6 a^2 \sin ^{-1}(a x)^2}+\frac {x^2}{3 \sin ^{-1}(a x)^2}+\frac {2 x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac {x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac {1}{6 a^2 \sin ^{-1}(a x)^2}+\frac {x^2}{3 \sin ^{-1}(a x)^2}+\frac {2 x \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)}-\frac {2 \text {Ci}\left (2 \sin ^{-1}(a x)\right )}{3 a^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 86, normalized size = 0.89 \[ \frac {-2 a x \sqrt {1-a^2 x^2}+4 a x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2+\left (2 a^2 x^2-1\right ) \sin ^{-1}(a x)-4 \sin ^{-1}(a x)^3 \text {Ci}\left (2 \sin ^{-1}(a x)\right )}{6 a^2 \sin ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\arcsin \left (a x\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 92, normalized size = 0.95 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x}{3 \, a \arcsin \left (a x\right )} - \frac {2 \, \operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{3 \, a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{3 \, a \arcsin \left (a x\right )^{3}} + \frac {a^{2} x^{2} - 1}{3 \, a^{2} \arcsin \left (a x\right )^{2}} + \frac {1}{6 \, a^{2} \arcsin \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 0.62 \[ \frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}-\frac {2 \Ci \left (2 \arcsin \left (a x \right )\right )}{3}}{a^{2}} \]
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 \[ -\frac {4 \, a^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3} \int \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{{\left (a^{3} x^{2} - a\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} - 2 \, {\left (2 \, a x \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} - a x\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - {\left (2 \, a^{2} x^{2} - 1\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{6 \, a^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}} \]
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 \[ \int \frac {x}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \]
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 \[ \int \frac {x}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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