Integrand size = 25, antiderivative size = 122 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2} \]
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Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4794, 4770, 4504, 4268, 2317, 2438, 197} \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {2 \text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{d^2}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}} \]
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Rule 197
Rule 2317
Rule 2438
Rule 4268
Rule 4504
Rule 4770
Rule 4794
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}(\int (a+b x) \csc (x) \sec (x) \, dx,x,\arccos (c x))}{d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 \text {Subst}(\int (a+b x) \csc (2 x) \, dx,x,\arccos (c x))}{d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{d^2}-\frac {b \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.91 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2}}{1-c x}-\frac {b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \arccos (c x)}{1-c x}+\frac {b \arccos (c x)}{1+c x}-4 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-4 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+4 b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )+4 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+4 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{4 d^2} \]
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Time = 2.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.09
method | result | size |
parts | \(\frac {a \left (\ln \left (x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}-\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\) | \(255\) |
derivativedivides | \(\frac {a \left (\ln \left (c x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}-\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\) | \(257\) |
default | \(\frac {a \left (\ln \left (c x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}-\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\) | \(257\) |
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\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]
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\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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