Integrand size = 18, antiderivative size = 53 \[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=\frac {2 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {\operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c} \]
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Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5903, 4267, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=\frac {2 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {\operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c} \]
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Rule 2317
Rule 2438
Rule 4267
Rule 5903
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arccosh}(a x))}{a c} \\ & = \frac {2 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c}-\frac {\text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {\operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.45 \[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=-\frac {\text {arccosh}(a x) \log \left (1-e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\text {arccosh}(a x) \log \left (1+e^{\text {arccosh}(a x)}\right )}{a c}+\frac {\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {\operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c} \]
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Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 169, normalized size of antiderivative = 3.19
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {arccosh}\left (a x \right )}{c}+\frac {2 i \left (\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \sqrt {-a^{2} x^{2}+1}\, \sqrt {\frac {a x}{2}+\frac {1}{2}}\, \sqrt {\frac {a x}{2}-\frac {1}{2}}}{c \left (a^{2} x^{2}-1\right )}}{a}\) | \(169\) |
default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {arccosh}\left (a x \right )}{c}+\frac {2 i \left (\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \sqrt {-a^{2} x^{2}+1}\, \sqrt {\frac {a x}{2}+\frac {1}{2}}\, \sqrt {\frac {a x}{2}-\frac {1}{2}}}{c \left (a^{2} x^{2}-1\right )}}{a}\) | \(169\) |
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\[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{a^{2} c x^{2} - c} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=- \frac {\int \frac {\operatorname {acosh}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx}{c} \]
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\[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{a^{2} c x^{2} - c} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{a^{2} c x^{2} - c} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)}{c-a^2 c x^2} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{c-a^2\,c\,x^2} \,d x \]
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