Integrand size = 25, antiderivative size = 177 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d^2}-\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2} \]
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Time = 0.17 (sec) , antiderivative size = 177, 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 = {5934, 5938, 5903, 4267, 2317, 2438, 75, 100, 21} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{2 c^5 d^2}-\frac {b x^2}{2 c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 21
Rule 75
Rule 100
Rule 2317
Rule 2438
Rule 4267
Rule 5903
Rule 5934
Rule 5938
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{2 c^2 d} \\ & = -\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \int \frac {x (-2-2 c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^3 d^2}-\frac {(3 b) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^3 d^2}-\frac {3 \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d} \\ & = -\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{2 c^5 d^2}+\frac {b \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^3 d^2} \\ & = -\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d^2}-\frac {(3 b) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 c^5 d^2}+\frac {(3 b) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 c^5 d^2} \\ & = -\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d^2}-\frac {(3 b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2} \\ & = -\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d^2}-\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {4 a c x-3 b \sqrt {\frac {-1+c x}{1+c x}}-4 b c x \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a c x}{-1+c^2 x^2}+4 b c x \text {arccosh}(c x)+\frac {b \text {arccosh}(c x)}{1-c x}-\frac {b \text {arccosh}(c x)}{1+c x}+6 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-6 b \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+3 a \log (1-c x)-3 a \log (1+c x)-6 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+6 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c^5 d^2} \]
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Time = 0.76 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c^{5}}\) | \(254\) |
default | \(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c^{5}}\) | \(254\) |
parts | \(\frac {a \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}\right )}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{d^{2} c^{4}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{c^{5} d^{2}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2} c^{5}}+\frac {3 b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}-\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2} c^{5}}-\frac {3 b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}\) | \(286\) |
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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