Optimal. Leaf size=57 \[ -\frac {b \sqrt {1-\frac {c^2}{x^2}}}{4 c x}+\frac {b \csc ^{-1}\left (\frac {x}{c}\right )}{4 c^2}-\frac {a+b \text {ArcSin}\left (\frac {c}{x}\right )}{2 x^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4926, 12, 342,
327, 222} \begin {gather*} -\frac {a+b \text {ArcSin}\left (\frac {c}{x}\right )}{2 x^2}-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{4 c x}+\frac {b \csc ^{-1}\left (\frac {x}{c}\right )}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 327
Rule 342
Rule 4926
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{x^3} \, dx &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{2 x^2}-\frac {1}{2} b \int \frac {c}{\sqrt {1-\frac {c^2}{x^2}} x^4} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{2 x^2}-\frac {1}{2} (b c) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}} x^4} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{2 x^2}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{4 c x}-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{2 x^2}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x^2}} \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{4 c x}+\frac {b \csc ^{-1}\left (\frac {x}{c}\right )}{4 c^2}-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 65, normalized size = 1.14 \begin {gather*} -\frac {a}{2 x^2}-\frac {b \sqrt {\frac {-c^2+x^2}{x^2}}}{4 c x}+\frac {b \text {ArcSin}\left (\frac {c}{x}\right )}{4 c^2}-\frac {b \text {ArcSin}\left (\frac {c}{x}\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 59, normalized size = 1.04
method | result | size |
derivativedivides | \(-\frac {\frac {a \,c^{2}}{2 x^{2}}+b \left (\frac {c^{2} \arcsin \left (\frac {c}{x}\right )}{2 x^{2}}+\frac {c \sqrt {1-\frac {c^{2}}{x^{2}}}}{4 x}-\frac {\arcsin \left (\frac {c}{x}\right )}{4}\right )}{c^{2}}\) | \(59\) |
default | \(-\frac {\frac {a \,c^{2}}{2 x^{2}}+b \left (\frac {c^{2} \arcsin \left (\frac {c}{x}\right )}{2 x^{2}}+\frac {c \sqrt {1-\frac {c^{2}}{x^{2}}}}{4 x}-\frac {\arcsin \left (\frac {c}{x}\right )}{4}\right )}{c^{2}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 86, normalized size = 1.51 \begin {gather*} \frac {1}{4} \, {\left (c {\left (\frac {x \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {c^{2}}{x^{2}} - 1\right )} - c^{4}} - \frac {\arctan \left (\frac {x \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c}\right )}{c^{3}}\right )} - \frac {2 \, \arcsin \left (\frac {c}{x}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.14, size = 55, normalized size = 0.96 \begin {gather*} -\frac {b c x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} + 2 \, a c^{2} + {\left (2 \, b c^{2} - b x^{2}\right )} \arcsin \left (\frac {c}{x}\right )}{4 \, c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.99, size = 112, normalized size = 1.96 \begin {gather*} - \frac {a}{2 x^{2}} - \frac {b c \left (\begin {cases} \frac {i \sqrt {\frac {c^{2}}{x^{2}} - 1}}{2 c^{2} x} + \frac {i \operatorname {acosh}{\left (\frac {c}{x} \right )}}{2 c^{3}} & \text {for}\: \left |{\frac {c^{2}}{x^{2}}}\right | > 1 \\- \frac {1}{2 x^{3} \sqrt {- \frac {c^{2}}{x^{2}} + 1}} + \frac {1}{2 c^{2} x \sqrt {- \frac {c^{2}}{x^{2}} + 1}} - \frac {\operatorname {asin}{\left (\frac {c}{x} \right )}}{2 c^{3}} & \text {otherwise} \end {cases}\right )}{2} - \frac {b \operatorname {asin}{\left (\frac {c}{x} \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 70, normalized size = 1.23 \begin {gather*} -\frac {\frac {2 \, b {\left (\frac {c^{2}}{x^{2}} - 1\right )} \arcsin \left (\frac {c}{x}\right )}{c} + \frac {2 \, a {\left (\frac {c^{2}}{x^{2}} - 1\right )}}{c} + \frac {b \arcsin \left (\frac {c}{x}\right )}{c} + \frac {b \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{x}}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 50, normalized size = 0.88 \begin {gather*} -\frac {a}{2\,x^2}-\frac {b\,\sqrt {1-\frac {c^2}{x^2}}}{4\,c\,x}-\frac {b\,\mathrm {asin}\left (\frac {c}{x}\right )\,\left (\frac {2\,c^2}{x^2}-1\right )}{4\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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