Integrand size = 25, antiderivative size = 180 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {3 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c^5 d^2} \]
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Time = 0.17 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4792, 4796, 4750, 4268, 2317, 2438, 267, 272, 45} \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {3 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c^5 d^2}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}} \]
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Rule 45
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4268
Rule 4750
Rule 4792
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx}{2 c^2 d} \\ & = \frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {(3 b) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d^2}+\frac {b \text {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac {3 \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx}{2 c^4 d} \\ & = -\frac {3 b \sqrt {1-c^2 x^2}}{2 c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arccos (c x))}{2 c^5 d^2}+\frac {b \text {Subst}\left (\int \left (\frac {1}{c^2 \left (1-c^2 x\right )^{3/2}}-\frac {1}{c^2 \sqrt {1-c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2} \\ & = \frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^5 d^2}-\frac {(3 b) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{2 c^5 d^2}+\frac {(3 b) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{2 c^5 d^2} \\ & = \frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{2 c^5 d^2}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{2 c^5 d^2} \\ & = \frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {3 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c^5 d^2} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.63 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a x}{c^4 d^2}-\frac {a x}{2 c^4 d^2 \left (-1+c^2 x^2\right )}+\frac {3 a \log (1-c x)}{4 c^5 d^2}-\frac {3 a \log (1+c x)}{4 c^5 d^2}+\frac {b \left (\frac {\sqrt {1-c^2 x^2}-\arccos (c x)}{4 c^4 \left (c+c^2 x\right )}+\frac {\sqrt {1-c^2 x^2}+\arccos (c x)}{4 c^4 \left (c-c^2 x\right )}+\frac {-\sqrt {1-c^2 x^2}+c x \arccos (c x)}{c^5}-\frac {3 \left (-\frac {i \arccos (c x)^2}{2 c}+\frac {2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}\right )}{4 c^4}-\frac {3 i \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1-e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{8 c^5}\right )}{d^2} \]
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Time = 2.41 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b \arccos \left (c x \right ) c x}{d^{2}}-\frac {b \arccos \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}}{c^{5}}\) | \(250\) |
default | \(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b \arccos \left (c x \right ) c x}{d^{2}}-\frac {b \arccos \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}}{c^{5}}\) | \(250\) |
parts | \(\frac {a \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}\right )}{d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{c^{5} d^{2}}+\frac {b x \arccos \left (c x \right )}{d^{2} c^{4}}-\frac {b x \arccos \left (c x \right )}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2} c^{5}}+\frac {3 i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2} c^{5}}-\frac {3 i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}\) | \(282\) |
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\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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