Optimal. Leaf size=177 \[ \frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 i b c \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {3 i b c \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 d^2} \]
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Rubi [A] time = 0.18, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4702, 4656, 4658, 4183, 2279, 2391, 261, 266, 51, 63, 208} \[ -\frac {3 i b c \text {PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {3 i b c \text {PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 261
Rule 266
Rule 2279
Rule 2391
Rule 4183
Rule 4656
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 )^2} \, dx &=-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\left (3 c^2\right ) \int \frac {a+b \cos ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (3 b c^3\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\left (3 c^2\right ) \int \frac {a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(3 c) \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{2 d^2}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d^2}\\ &=\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{d^2}+\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^2}+\frac {(3 b c) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 d^2}-\frac {(3 b c) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 d^2}\\ &=\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{d^2}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 d^2}\\ &=\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{d^2}+\frac {b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 i b c \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {3 i b c \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 251, normalized size = 1.42 \[ \frac {-\frac {2 a c^2 x}{c^2 x^2-1}-3 a c \log (1-c x)+3 a c \log (c x+1)-\frac {4 a}{x}+\frac {b c \sqrt {1-c^2 x^2}}{1-c x}+\frac {b c \sqrt {1-c^2 x^2}}{c x+1}+4 b c \log \left (\sqrt {1-c^2 x^2}+1\right )-6 i b c \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )+6 i b c \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )-4 b c \log (x)+\frac {b c \cos ^{-1}(c x)}{1-c x}-\frac {b c \cos ^{-1}(c x)}{c x+1}-\frac {4 b \cos ^{-1}(c x)}{x}-6 b c \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+6 b c \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{4 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arccos \left (c x\right ) + a}{c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 260, normalized size = 1.47 \[ -\frac {c a}{4 d^{2} \left (c x +1\right )}+\frac {3 c a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{d^{2} x}-\frac {c a}{4 d^{2} \left (c x -1\right )}-\frac {3 c a \ln \left (c x -1\right )}{4 d^{2}}-\frac {3 b \arccos \left (c x \right ) c^{2} x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {c b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \arccos \left (c x \right )}{d^{2} x \left (c^{2} x^{2}-1\right )}-\frac {2 i c b \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 i c b \dilog \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 c b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i c b \dilog \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}} \]
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 \[ -\frac {1}{4} \, a {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} - 2\right )}}{c^{2} d^{2} x^{3} - d^{2} x} - \frac {3 \, c \log \left (c x + 1\right )}{d^{2}} + \frac {3 \, c \log \left (c x - 1\right )}{d^{2}}\right )} - \frac {{\left ({\left (6 \, c^{2} x^{2} - 3 \, {\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{3} x^{3} - c x\right )} \log \left (-c x + 1\right ) - 4\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) - {\left (c^{2} d^{2} x^{3} - d^{2} x\right )} \int \frac {{\left (6 \, c^{3} x^{2} - 3 \, {\left (c^{4} x^{3} - c^{2} x\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{4} x^{3} - c^{2} x\right )} \log \left (-c x + 1\right ) - 4 \, c\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x}\right )} b}{4 \, {\left (c^{2} d^{2} x^{3} - d^{2} x\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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