Integrand size = 25, antiderivative size = 140 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d} \]
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Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5938, 5913, 3797, 2221, 2317, 2438, 92, 54} \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}-\frac {\log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}+\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{4 c^3 d} \]
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Rule 54
Rule 92
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5913
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {\int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{c^2}+\frac {b \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c d} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}-\frac {\text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{c^4 d}+\frac {b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^3 d} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{c^4 d} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^4 d} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=-\frac {2 c^2 x^2 (a+b \text {arccosh}(c x))-\frac {2 (a+b \text {arccosh}(c x))^2}{b}-b \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+4 (a+b \text {arccosh}(c x)) \log \left (1-e^{\text {arccosh}(c x)}\right )+4 (a+b \text {arccosh}(c x)) \log \left (1+e^{\text {arccosh}(c x)}\right )+4 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+4 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c^4 d} \]
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Time = 0.61 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.55
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\frac {c^{2} x^{2}}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (c x \right )}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{4}}\) | \(217\) |
default | \(\frac {-\frac {a \left (\frac {c^{2} x^{2}}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (c x \right )}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{4}}\) | \(217\) |
parts | \(-\frac {a \,x^{2}}{2 d \,c^{2}}-\frac {a \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{4}}+\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 d \,c^{4}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{2}}{2 d \,c^{2}}+\frac {b x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c^{3} d}+\frac {b \,\operatorname {arccosh}\left (c x \right )}{4 c^{4} d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}-\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}\) | \(233\) |
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\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x^{3}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{3} \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
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\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{c^{2} d x^{2} - d} \,d x } \]
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Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
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