Optimal. Leaf size=31 \[ a x+b c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right )+b x \csc ^{-1}\left (\frac {x}{c}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4832, 5215, 266, 63, 208} \[ a x+b c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right )+b x \csc ^{-1}\left (\frac {x}{c}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 4832
Rule 5215
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (\frac {c}{x}\right ) \, dx\\ &=a x+b \int \csc ^{-1}\left (\frac {x}{c}\right ) \, dx\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+(b c) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}} x} \, dx\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-\frac {c^2}{x^2}}\right )}{c}\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+b c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.10, size = 89, normalized size = 2.87 \[ a x+\frac {b c \sqrt {x^2-c^2} \left (\log \left (\frac {x}{\sqrt {x^2-c^2}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2-c^2}}\right )\right )}{2 x \sqrt {1-\frac {c^2}{x^2}}}+b x \sin ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 75, normalized size = 2.42 \[ -b c \log \left (x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} - x\right ) + a x + {\left (b x - b\right )} \arcsin \left (\frac {c}{x}\right ) - 2 \, b \arctan \left (\frac {x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} - x}{c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.71, size = 60, normalized size = 1.94 \[ a x + \frac {{\left (c^{2} {\left (\log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )\right )} + 2 \, c x \arcsin \left (\frac {c}{x}\right )\right )} b}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 1.19 \[ a x -b c \left (-\frac {\arcsin \left (\frac {c}{x}\right ) x}{c}-\arctanh \left (\frac {1}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 52, normalized size = 1.68 \[ \frac {1}{2} \, {\left (c {\left (\log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right ) - \log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} - 1\right )\right )} + 2 \, x \arcsin \left (\frac {c}{x}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 32, normalized size = 1.03 \[ a\,x+b\,x\,\mathrm {asin}\left (\frac {c}{x}\right )+b\,c\,\mathrm {sign}\relax (x)\,\ln \left (x+\sqrt {x^2-c^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.70, size = 32, normalized size = 1.03 \[ a x + b \left (c \left (\begin {cases} \operatorname {acosh}{\left (\frac {x}{c} \right )} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {x}{c} \right )} & \text {otherwise} \end {cases}\right ) + x \operatorname {asin}{\left (\frac {c}{x} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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