Optimal. Leaf size=39 \[ \frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5221, 191} \[ \frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5221
Rubi steps
\begin {align*} \int x \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{2 c}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.28 \[ \frac {a x^2}{2}+\frac {b x \sqrt {\frac {c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac {1}{2} b x^2 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 39, normalized size = 1.00 \[ \frac {b c^{2} x^{2} \operatorname {arccsc}\left (c x\right ) + a c^{2} x^{2} + \sqrt {c^{2} x^{2} - 1} b}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 182, normalized size = 4.67 \[ \frac {1}{8} \, {\left (\frac {b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {2 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {2 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {2 \, a}{c^{3}} - \frac {2 \, b}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {b \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {a}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 65, normalized size = 1.67 \[ \frac {\frac {c^{2} x^{2} a}{2}+b \left (\frac {c^{2} x^{2} \mathrm {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 36, normalized size = 0.92 \[ \frac {1}{2} \, a x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 40, normalized size = 1.03 \[ \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{2}+\frac {b\,x\,\sqrt {1-\frac {1}{c^2\,x^2}}}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.14, size = 58, normalized size = 1.49 \[ \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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