Integrand size = 25, antiderivative size = 177 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d^2}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 d^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 d^2} \]
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Time = 0.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4790, 4748, 4750, 4268, 2317, 2438, 267, 272, 53, 65, 214} \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {3 c \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{d^2}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {a+b \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}-\frac {3 i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 d^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 d^2}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}} \]
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Rule 53
Rule 65
Rule 214
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4268
Rule 4748
Rule 4750
Rule 4790
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\left (3 c^2\right ) \int \frac {a+b \arccos (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 \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \text {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 \arccos (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 \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(3 c) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arccos (c x))}{2 d^2}-\frac {(b c) \text {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 \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d^2}+\frac {b \text {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) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{2 d^2}-\frac {(3 b c) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{2 d^2} \\ & = \frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d^2}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {(3 i b c) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{2 d^2}+\frac {(3 i b c) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{2 d^2} \\ & = \frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d^2}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 d^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 d^2} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {4 a}{x}+\frac {b c \sqrt {1-c^2 x^2}}{1-c x}+\frac {b c \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a c^2 x}{-1+c^2 x^2}-\frac {4 b \arccos (c x)}{x}+\frac {b c \arccos (c x)}{1-c x}-\frac {b c \arccos (c x)}{1+c x}-6 b c \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+6 b c \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-4 b c \log (x)-3 a c \log (1-c x)+3 a c \log (1+c x)+4 b c \log \left (1+\sqrt {1-c^2 x^2}\right )-6 i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+6 i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{4 d^2} \]
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Time = 4.35 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.88
method | result | size |
parts | \(\frac {a \left (-\frac {1}{x}-\frac {c}{4 \left (c x -1\right )}-\frac {3 c \ln \left (c x -1\right )}{4}-\frac {c}{4 \left (c x +1\right )}+\frac {3 c \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {i b \left (3 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) c^{3} x^{3}-3 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) c x -3 i \arccos \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}+3 \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}+3 \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}-i c x \sqrt {-c^{2} x^{2}+1}+2 i \arccos \left (c x \right )-4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x -3 \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x -3 \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x}\) | \(332\) |
derivativedivides | \(c \left (\frac {a \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {i b \left (3 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) c^{3} x^{3}-3 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) c x -3 i \arccos \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}+3 \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}+3 \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}-i c x \sqrt {-c^{2} x^{2}+1}+2 i \arccos \left (c x \right )-4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x -3 \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x -3 \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) c x}\right )\) | \(336\) |
default | \(c \left (\frac {a \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {i b \left (3 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) c^{3} x^{3}-3 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) c x -3 i \arccos \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}+3 \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}+3 \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{3} x^{3}-i c x \sqrt {-c^{2} x^{2}+1}+2 i \arccos \left (c x \right )-4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x -3 \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x -3 \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) c x}\right )\) | \(336\) |
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\[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
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\[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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