Optimal. Leaf size=62 \[ -\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{b \left (1-\frac{c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3} \]
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Rubi [A] time = 0.0474299, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 266, 43} \[ -\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{b \left (1-\frac{c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{x^4} \, dx &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{3} b \int \frac{c}{\sqrt{1-\frac{c^2}{x^2}} x^5} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{3} (b c) \int \frac{1}{\sqrt{1-\frac{c^2}{x^2}} x^5} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3}+\frac{b \left (1-\frac{c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0553837, size = 60, normalized size = 0.97 \[ -\frac{a}{3 x^3}+b \left (-\frac{2}{9 c^3}-\frac{1}{9 c x^2}\right ) \sqrt{\frac{x^2-c^2}{x^2}}-\frac{b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 67, normalized size = 1.1 \begin{align*} -{\frac{1}{{c}^{3}} \left ({\frac{a{c}^{3}}{3\,{x}^{3}}}+b \left ({\frac{{c}^{3}}{3\,{x}^{3}}\arcsin \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{2}}{9\,{x}^{2}}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}}+{\frac{2}{9}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43452, size = 78, normalized size = 1.26 \begin{align*} \frac{1}{9} \,{\left (c{\left (\frac{{\left (-\frac{c^{2}}{x^{2}} + 1\right )}^{\frac{3}{2}}}{c^{4}} - \frac{3 \, \sqrt{-\frac{c^{2}}{x^{2}} + 1}}{c^{4}}\right )} - \frac{3 \, \arcsin \left (\frac{c}{x}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26782, size = 126, normalized size = 2.03 \begin{align*} -\frac{3 \, b c^{3} \arcsin \left (\frac{c}{x}\right ) + 3 \, a c^{3} +{\left (b c^{2} x + 2 \, b x^{3}\right )} \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}}}{9 \, c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.5698, size = 114, normalized size = 1.84 \begin{align*} - \frac{a}{3 x^{3}} - \frac{b c \left (\begin{cases} \frac{\sqrt{-1 + \frac{x^{2}}{c^{2}}}}{3 c x^{3}} + \frac{2 \sqrt{-1 + \frac{x^{2}}{c^{2}}}}{3 c^{3} x} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{c^{2}}\right |} > 1 \\\frac{i \sqrt{1 - \frac{x^{2}}{c^{2}}}}{3 c x^{3}} + \frac{2 i \sqrt{1 - \frac{x^{2}}{c^{2}}}}{3 c^{3} x} & \text{otherwise} \end{cases}\right )}{3} - \frac{b \operatorname{asin}{\left (\frac{c}{x} \right )}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (\frac{c}{x}\right ) + a}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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