Optimal. Leaf size=60 \[ -\frac{a+b \csc ^{-1}(c x)}{3 x^3}+\frac{1}{9} b c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{1}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \]
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Rubi [A] time = 0.0379044, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5221, 266, 43} \[ -\frac{a+b \csc ^{-1}(c x)}{3 x^3}+\frac{1}{9} b c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{1}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5221
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{x^4} \, dx &=-\frac{a+b \csc ^{-1}(c x)}{3 x^3}-\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^5} \, dx}{3 c}\\ &=-\frac{a+b \csc ^{-1}(c x)}{3 x^3}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=-\frac{a+b \csc ^{-1}(c x)}{3 x^3}+\frac{b \operatorname{Subst}\left (\int \left (\frac{c^2}{\sqrt{1-\frac{x}{c^2}}}-c^2 \sqrt{1-\frac{x}{c^2}}\right ) \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=-\frac{1}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}}+\frac{1}{9} b c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{a+b \csc ^{-1}(c x)}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0650652, size = 59, normalized size = 0.98 \[ -\frac{a}{3 x^3}+b \left (-\frac{2 c^3}{9}-\frac{c}{9 x^2}\right ) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.176, size = 75, normalized size = 1.3 \begin{align*}{c}^{3} \left ( -{\frac{a}{3\,{c}^{3}{x}^{3}}}+b \left ( -{\frac{{\rm arccsc} \left (cx\right )}{3\,{c}^{3}{x}^{3}}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{9\,{c}^{4}{x}^{4}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{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 = 0.977066, size = 78, normalized size = 1.3 \begin{align*} \frac{1}{9} \, b{\left (\frac{c^{4}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, c^{4} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{3 \, \operatorname{arccsc}\left (c x\right )}{x^{3}}\right )} - \frac{a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23779, size = 97, normalized size = 1.62 \begin{align*} -\frac{3 \, b \operatorname{arccsc}\left (c x\right ) +{\left (2 \, b c^{2} x^{2} + b\right )} \sqrt{c^{2} x^{2} - 1} + 3 \, a}{9 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acsc}{\left (c x \right )}}{x^{4}}\, dx \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 \operatorname{arccsc}\left (c x\right ) + a}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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