Integrand size = 22, antiderivative size = 120 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c d^2} \]
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Time = 0.09 (sec) , antiderivative size = 120, 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 = {5901, 5903, 4267, 2317, 2438, 75} \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c d^2}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {b}{2 c d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 75
Rule 2317
Rule 2438
Rule 4267
Rule 5901
Rule 5903
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{2 c d^2} \\ & = -\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 c d^2} \\ & = -\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 c d^2} \\ & = -\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c d^2} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.58 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {-2 a c x-2 b \sqrt {\frac {-1+c x}{1+c x}}-2 b c x \sqrt {\frac {-1+c x}{1+c x}}-2 b \text {arccosh}(c x) \left (c x+\left (-1+c^2 x^2\right ) \log \left (1-e^{\text {arccosh}(c x)}\right )+\left (1-c^2 x^2\right ) \log \left (1+e^{\text {arccosh}(c x)}\right )\right )+\left (a-a c^2 x^2\right ) \log (1-c x)-a \log (1+c x)+a c^2 x^2 \log (1+c x)}{-1+c^2 x^2}+2 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c d^2} \]
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Time = 0.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {\frac {a \left (-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\) | \(192\) |
default | \(\frac {\frac {a \left (-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\) | \(192\) |
parts | \(\frac {a \left (-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2} c}\) | \(203\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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