Integrand size = 14, antiderivative size = 352 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \]
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Time = 0.47 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5963, 5980, 3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=-\frac {c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^3 d \text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )} \]
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3401
Rule 3405
Rule 5963
Rule 5980
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c \int \frac {\text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{e} \\ & = -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \text {Subst}\left (\int \frac {x}{(c d+e \cosh (x))^2} \, dx,x,\text {arccosh}(c x)\right )}{e} \\ & = -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \text {Subst}\left (\int \frac {\sinh (x)}{c d+e \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{c^2 d^2-e^2}+\frac {\left (c^3 d\right ) \text {Subst}\left (\int \frac {x}{c d+e \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{e \left (c^2 d^2-e^2\right )} \\ & = -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \text {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 c^3 d\right ) \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 \left (c^2 d^2-e^2\right )} \\ & = -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 c^3 d\right ) \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 )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (2 c^3 d\right ) \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 )}{\left (c^2 d^2-e^2\right )^{3/2}} \\ & = -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (c^3 d\right ) \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 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (c^3 d\right ) \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 \left (c^2 d^2-e^2\right )^{3/2}} \\ & = -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (c^3 d\right ) \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 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (c^3 d\right ) \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 \left (c^2 d^2-e^2\right )^{3/2}} \\ & = -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.17 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.66 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{(c d-e) (c d+e) (c d+c e x)}-\frac {\text {arccosh}(c x)^2}{2 e (c d+c e x)^2}+\frac {\log \left (1+\frac {e x}{d}\right )}{c^2 d^2 e-e^3}+\frac {c d \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 )}{e \left (-c^2 d^2+e^2\right )^{3/2}}\right ) \]
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Time = 0.74 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.72
method | result | size |
derivativedivides | \(\frac {-\frac {c^{3} \operatorname {arccosh}\left (c x \right ) \left (c^{2} d^{2} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c d e +2 \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} c x -2 c^{2} d^{2}-4 d \,c^{2} e x -2 e^{2} c^{2} x^{2}-e^{2} \operatorname {arccosh}\left (c x \right )\right )}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2}}+\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} d \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}+\frac {c^{3} \ln \left (2 d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}}{c}\) | \(605\) |
default | \(\frac {-\frac {c^{3} \operatorname {arccosh}\left (c x \right ) \left (c^{2} d^{2} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c d e +2 \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} c x -2 c^{2} d^{2}-4 d \,c^{2} e x -2 e^{2} c^{2} x^{2}-e^{2} \operatorname {arccosh}\left (c x \right )\right )}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2}}+\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} d \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}+\frac {c^{3} \ln \left (2 d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}}{c}\) | \(605\) |
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\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int \frac {{\mathrm {acosh}\left (c\,x\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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