Integrand size = 22, antiderivative size = 132 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c d^2} \]
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Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4748, 4750, 4268, 2317, 2438, 267} \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c d^2}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c d^2}+\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}} \]
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Rule 267
Rule 2317
Rule 2438
Rule 4268
Rule 4748
Rule 4750
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx}{2 d} \\ & = \frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arccos (c x))}{2 c d^2} \\ & = \frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c d^2}+\frac {b \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{2 c d^2} \\ & = \frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c d^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{2 c d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{2 c d^2} \\ & = \frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c d^2} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.67 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {b \sqrt {1-c^2 x^2}}{c+c^2 x}-\frac {2 a x}{-1+c^2 x^2}+\frac {b \arccos (c x)}{c-c^2 x}-\frac {b \arccos (c x)}{c+c^2 x}-\frac {2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )}{c}+\frac {2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )}{c}-\frac {a \log (1-c x)}{c}+\frac {a \log (1+c x)}{c}-\frac {2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}+\frac {2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c}}{4 d^2} \]
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Time = 1.86 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.43
method | result | size |
derivativedivides | \(\frac {\frac {a \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}}{c}\) | \(189\) |
default | \(\frac {\frac {a \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}}{c}\) | \(189\) |
parts | \(\frac {a \left (-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}\right )}{d^{2}}+\frac {b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c}\) | \(200\) |
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\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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