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 = 0.0218781, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 4832
Rule 5215
Rule 266
Rule 63
Rule 208
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*}
Mathematica [B] time = 0.089225, 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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Maple [A] time = 0.005, size = 37, normalized size = 1.2 \begin{align*} ax-bc \left ( -{\frac{x}{c}\arcsin \left ({\frac{c}{x}} \right ) }-{\it Artanh} \left ({\frac{1}{\sqrt{1-{\frac{{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 = 1.41117, size = 70, normalized size = 2.26 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.48798, size = 158, normalized size = 5.1 \begin{align*} -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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.50845, size = 34, normalized size = 1.1 \begin{align*} a x + b \left (c \left (\begin{cases} \operatorname{acosh}{\left (\frac{x}{c} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17106, size = 68, normalized size = 2.19 \begin{align*} \frac{1}{2} \,{\left ({\left (\log \left (c^{2}\right ) \mathrm{sgn}\left (x\right ) - \frac{2 \, \log \left ({\left | -x + \sqrt{-c^{2} + x^{2}} \right |}\right )}{\mathrm{sgn}\left (x\right )}\right )} c + 2 \, x \arcsin \left (\frac{c}{x}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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