Integrand size = 14, antiderivative size = 259 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=-\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]
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Time = 0.38 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5963, 5980, 3401, 2296, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 c \text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\text {arccosh}(c x)^2}{e (d+e x)} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3401
Rule 5963
Rule 5980
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {(2 c) \int \frac {\text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{e} \\ & = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {(2 c) \text {Subst}\left (\int \frac {x}{c d+e \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{e} \\ & = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {(4 c) \text {Subst}\left (\int \frac {e^x x}{e+2 c d e^x+e e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{e} \\ & = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {(4 c) \text {Subst}\left (\int \frac {e^x x}{2 c d-2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {c^2 d^2-e^2}}-\frac {(4 c) \text {Subst}\left (\int \frac {e^x x}{2 c d+2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {c^2 d^2-e^2}} \\ & = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {(2 c) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {(2 c) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{e \sqrt {c^2 d^2-e^2}} \\ & = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {(2 c) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {(2 c) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{e \sqrt {c^2 d^2-e^2}} \\ & = -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.39 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.27 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=-\frac {c \left (\frac {\text {arccosh}(c x)^2}{c d+c e x}+\frac {2 \left (2 \text {arccosh}(c x) \arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )-2 i \arccos \left (-\frac {c d}{e}\right ) \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\left (\arccos \left (-\frac {c d}{e}\right )+2 \left (\arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{-\frac {1}{2} \text {arccosh}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )+\left (\arccos \left (-\frac {c d}{e}\right )-2 \left (\arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{\frac {1}{2} \text {arccosh}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )-\left (\arccos \left (-\frac {c d}{e}\right )+2 \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {-c^2 d^2+e^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\left (\arccos \left (-\frac {c d}{e}\right )-2 \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {-c^2 d^2+e^2}\right ) \left (1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c d-i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c d+i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )\right )\right )}{\sqrt {-c^2 d^2+e^2}}\right )}{e} \]
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Time = 0.70 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.50
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {arccosh}\left (c x \right )^{2} c^{2}}{e \left (e c x +c d \right )}+\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}}{c}\) | \(388\) |
default | \(\frac {-\frac {\operatorname {arccosh}\left (c x \right )^{2} c^{2}}{e \left (e c x +c d \right )}+\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}}{c}\) | \(388\) |
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\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\int \frac {{\mathrm {acosh}\left (c\,x\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]
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