Optimal. Leaf size=107 \[ -\frac {a+b \cos ^{-1}(c x)}{d x}+\frac {2 c \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d}-\frac {i b c \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{d}+\frac {i b c \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4702, 4658, 4183, 2279, 2391, 266, 63, 208} \[ -\frac {i b c \text {PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{d}+\frac {i b c \text {PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{d}-\frac {a+b \cos ^{-1}(c x)}{d x}+\frac {2 c \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 2279
Rule 2391
Rule 4183
Rule 4658
Rule 4702
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx &=-\frac {a+b \cos ^{-1}(c x)}{d x}+c^2 \int \frac {a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx-\frac {(b c) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {a+b \cos ^{-1}(c x)}{d x}-\frac {c \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{d}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {a+b \cos ^{-1}(c x)}{d x}+\frac {2 c \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{d}+\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 d}+\frac {(b c) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d}-\frac {(b c) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=-\frac {a+b \cos ^{-1}(c x)}{d x}+\frac {2 c \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{d}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d}-\frac {(i b c) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{d}+\frac {(i b c) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{d}\\ &=-\frac {a+b \cos ^{-1}(c x)}{d x}+\frac {2 c \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{d}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d}-\frac {i b c \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{d}+\frac {i b c \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 158, normalized size = 1.48 \[ -\frac {a c x \log (1-c x)-a c x \log (c x+1)+2 a-2 b c x \log \left (\sqrt {1-c^2 x^2}+1\right )+2 i b c x \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )-2 i b c x \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )+2 b c x \log (x)+2 b \cos ^{-1}(c x)+2 b c x \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )-2 b c x \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{2 d x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \arccos \left (c x\right ) + a}{c^{2} d x^{4} - d x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 166, normalized size = 1.55 \[ \frac {c a \ln \left (c x +1\right )}{2 d}-\frac {a}{d x}-\frac {c a \ln \left (c x -1\right )}{2 d}-\frac {b \arccos \left (c x \right )}{d x}-\frac {2 i c b \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i c b \dilog \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {c b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i c b \dilog \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {c \log \left (c x + 1\right )}{d} - \frac {c \log \left (c x - 1\right )}{d} - \frac {2}{d x}\right )} - \frac {{\left (d x \int \frac {{\left (c^{2} x \log \left (c x + 1\right ) - c^{2} x \log \left (-c x + 1\right ) - 2 \, c\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{2} d x^{3} - d x}\,{d x} - {\left (c x \log \left (c x + 1\right ) - c x \log \left (-c x + 1\right ) - 2\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )\right )} b}{2 \, d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^2\,\left (d-c^2\,d\,x^2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {a}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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