Integrand size = 27, antiderivative size = 151 \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5946, 4265, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]
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Rule 2317
Rule 2438
Rule 4265
Rule 5946
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \\ & = \frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {a \log (x)}{\sqrt {d}}-\frac {a \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}-\frac {i b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x) \left (\log \left (1-i e^{-\text {arccosh}(c x)}\right )-\log \left (1+i e^{-\text {arccosh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}} \]
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Time = 0.90 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.16
method | result | size |
default | \(-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+b \left (\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\right )\) | \(326\) |
parts | \(-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+b \left (\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\right )\) | \(326\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,\sqrt {d-c^2\,d\,x^2}} \,d x \]
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