Optimal. Leaf size=33 \[ -\frac{a+b \sin ^{-1}(c x)}{x}-b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]
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Rubi [A] time = 0.0265272, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {4627, 266, 63, 208} \[ -\frac{a+b \sin ^{-1}(c x)}{x}-b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 4627
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^2} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{x}+(b c) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{a+b \sin ^{-1}(c x)}{x}+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a+b \sin ^{-1}(c x)}{x}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c}\\ &=-\frac{a+b \sin ^{-1}(c x)}{x}-b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0027337, size = 36, normalized size = 1.09 \[ -\frac{a}{x}-b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-\frac{b \sin ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 43, normalized size = 1.3 \begin{align*} c \left ( -{\frac{a}{cx}}+b \left ( -{\frac{\arcsin \left ( cx \right ) }{cx}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50488, size = 63, normalized size = 1.91 \begin{align*} -{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} b - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55218, size = 140, normalized size = 4.24 \begin{align*} -\frac{b c x \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) - b c x \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) + 2 \, b \arcsin \left (c x\right ) + 2 \, a}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.3225, size = 39, normalized size = 1.18 \begin{align*} - \frac{a}{x} + b c \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{c x} \right )} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{c x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{b \operatorname{asin}{\left (c x \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.58261, size = 439, normalized size = 13.3 \begin{align*} -\frac{\sqrt{-c^{2} x^{2} + 1} b c^{2} x \arcsin \left (c x\right )}{2 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac{b c^{2} x \arcsin \left (c x\right )}{2 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac{\sqrt{-c^{2} x^{2} + 1} a c^{2} x}{2 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b c \log \left ({\left | c \right |}{\left | x \right |}\right )}{\sqrt{-c^{2} x^{2} + 1} + 1} - \frac{\sqrt{-c^{2} x^{2} + 1} b c \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}{\sqrt{-c^{2} x^{2} + 1} + 1} - \frac{a c^{2} x}{2 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac{b c \log \left ({\left | c \right |}{\left | x \right |}\right )}{\sqrt{-c^{2} x^{2} + 1} + 1} - \frac{b c \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}{\sqrt{-c^{2} x^{2} + 1} + 1} - \frac{\sqrt{-c^{2} x^{2} + 1} b \arcsin \left (c x\right )}{2 \, x} - \frac{b \arcsin \left (c x\right )}{2 \, x} - \frac{\sqrt{-c^{2} x^{2} + 1} a}{2 \, x} - \frac{a}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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