Optimal. Leaf size=51 \[ -\frac {a+b \csc ^{-1}(c x)}{2 x^2}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \csc ^{-1}(c x) \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5221, 335, 321, 216} \[ -\frac {a+b \csc ^{-1}(c x)}{2 x^2}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 335
Rule 5221
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^3} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{2 x^2}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^4} \, dx}{2 c}\\ &=-\frac {a+b \csc ^{-1}(c x)}{2 x^2}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {a+b \csc ^{-1}(c x)}{2 x^2}+\frac {1}{4} (b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \csc ^{-1}(c x)-\frac {a+b \csc ^{-1}(c x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 1.29 \[ -\frac {a}{2 x^2}-\frac {b c \sqrt {\frac {c^2 x^2-1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 \sin ^{-1}\left (\frac {1}{c x}\right )-\frac {b \csc ^{-1}(c x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 40, normalized size = 0.78 \[ \frac {{\left (b c^{2} x^{2} - 2 \, b\right )} \operatorname {arccsc}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1} b - 2 \, a}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 66, normalized size = 1.29 \[ -\frac {1}{4} \, {\left (2 \, b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 2 \, a c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + b c \arcsin \left (\frac {1}{c x}\right ) + \frac {b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 118, normalized size = 2.31 \[ -\frac {a}{2 x^{2}}-\frac {b \,\mathrm {arccsc}\left (c x \right )}{2 x^{2}}+\frac {c b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {c b}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b}{4 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 83, normalized size = 1.63 \[ \frac {1}{4} \, b {\left (\frac {\frac {c^{4} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1} - c^{3} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}{c} - \frac {2 \, \operatorname {arccsc}\left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 50, normalized size = 0.98 \[ -\frac {a}{2\,x^2}-\frac {b\,c^2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\left (\frac {2}{c^2\,x^2}-1\right )}{4}-\frac {b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}}{4\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.36, size = 121, normalized size = 2.37 \[ - \frac {a}{2 x^{2}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{2 x^{2}} - \frac {b \left (\begin {cases} \frac {i c^{3} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {i c^{2} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {c^{3} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} + \frac {c^{2}}{2 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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