Integrand size = 14, antiderivative size = 481 \[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Time = 0.58 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5909, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}+1\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}+1\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5909
Rule 5962
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-c} \text {arccosh}(a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \text {arccosh}(a x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}} \\ & = -\frac {\text {Subst}\left (\int \frac {x \sinh (x)}{a \sqrt {-c}-\sqrt {d} \cosh (x)} \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c}}-\frac {\text {Subst}\left (\int \frac {x \sinh (x)}{a \sqrt {-c}+\sqrt {d} \cosh (x)} \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c}} \\ & = -\frac {\text {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}-\sqrt {-a^2 c-d}-\sqrt {d} e^x} \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c}}-\frac {\text {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}+\sqrt {-a^2 c-d}-\sqrt {d} e^x} \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c}}-\frac {\text {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}-\sqrt {-a^2 c-d}+\sqrt {d} e^x} \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c}}-\frac {\text {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}+\sqrt {-a^2 c-d}+\sqrt {d} e^x} \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c}} \\ & = \frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1-\frac {\sqrt {d} e^x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1+\frac {\sqrt {d} e^x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1-\frac {\sqrt {d} e^x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1+\frac {\sqrt {d} e^x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {-c} \sqrt {d}} \\ & = \frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}} \\ & = \frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\frac {-\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )+\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )-\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )-\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 31.54 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.46
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{2}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{2}}{a}\) | \(222\) |
default | \(\frac {\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{2}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{2}}{a}\) | \(222\) |
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\[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{c + d x^{2}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{d\,x^2+c} \,d x \]
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