Optimal. Leaf size=32 \[ b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {a+b \cos ^{-1}(c x)}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {4628, 266, 63, 208} \[ b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {a+b \cos ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 4628
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \cos ^{-1}(c x)}{x}-(b c) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {a+b \cos ^{-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 \cos ^{-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 \cos ^{-1}(c x)}{x}+b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.34 \[ -\frac {a}{x}+b c \log \left (\sqrt {1-c^2 x^2}+1\right )-b c \log (x)-\frac {b \cos ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 92, normalized size = 2.88 \[ \frac {b c x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) - 2 \, b x \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, {\left (b x - b\right )} \arccos \left (c x\right ) - 2 \, a}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 347, normalized size = 10.84 \[ -\frac {b c \arccos \left (c x\right )}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} + \frac {b c \log \left ({\left | c x + \sqrt {-c^{2} x^{2} + 1} + 1 \right |}\right )}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} - \frac {b c \log \left ({\left | -c x + \sqrt {-c^{2} x^{2} + 1} - 1 \right |}\right )}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} - \frac {a c}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} + \frac {{\left (c^{2} x^{2} - 1\right )} b c \arccos \left (c x\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} b c \log \left ({\left | c x + \sqrt {-c^{2} x^{2} + 1} + 1 \right |}\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} - \frac {{\left (c^{2} x^{2} - 1\right )} b c \log \left ({\left | -c x + \sqrt {-c^{2} x^{2} + 1} - 1 \right |}\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} a c}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 1.28 \[ c \left (-\frac {a}{c x}+b \left (-\frac {\arccos \left (c x \right )}{c x}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 47, normalized size = 1.47 \[ {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arccos \left (c x\right )}{x}\right )} b - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 33, normalized size = 1.03 \[ b\,c\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,\mathrm {acos}\left (c\,x\right )}{x}-\frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.63, size = 41, normalized size = 1.28 \[ - \frac {a}{x} - b c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b \operatorname {acos}{\left (c x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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