Integrand size = 14, antiderivative size = 272 \[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=-\frac {\text {arccosh}(c x)^3}{3 e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \]
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Time = 0.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5962, 5681, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\text {arccosh}(c x)^3}{3 e} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 5681
Rule 5962
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^2 \sinh (x)}{c d+e \cosh (x)} \, dx,x,\text {arccosh}(c x)\right ) \\ & = -\frac {\text {arccosh}(c x)^3}{3 e}+\text {Subst}\left (\int \frac {e^x x^2}{c d-\sqrt {c^2 d^2-e^2}+e e^x} \, dx,x,\text {arccosh}(c x)\right )+\text {Subst}\left (\int \frac {e^x x^2}{c d+\sqrt {c^2 d^2-e^2}+e e^x} \, dx,x,\text {arccosh}(c x)\right ) \\ & = -\frac {\text {arccosh}(c x)^3}{3 e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Subst}\left (\int x \log \left (1+\frac {e e^x}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{e}-\frac {2 \text {Subst}\left (\int x \log \left (1+\frac {e e^x}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{e} \\ & = -\frac {\text {arccosh}(c x)^3}{3 e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e e^x}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{e}-\frac {2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e e^x}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{e} \\ & = -\frac {\text {arccosh}(c x)^3}{3 e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {e x}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{e}-\frac {2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{e} \\ & = -\frac {\text {arccosh}(c x)^3}{3 e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=-\frac {\text {arccosh}(c x)^3-3 \text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-3 \text {arccosh}(c x)^2 \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-6 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,\frac {e e^{\text {arccosh}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-6 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+6 \operatorname {PolyLog}\left (3,\frac {e e^{\text {arccosh}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+6 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{3 e} \]
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\[\int \frac {\operatorname {arccosh}\left (c x \right )^{2}}{e x +d}d x\]
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\[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{e x + d} \,d x } \]
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\[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (c x \right )}}{d + e x}\, dx \]
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\[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{e x + d} \,d x } \]
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\[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(c x)^2}{d+e x} \, dx=\int \frac {{\mathrm {acosh}\left (c\,x\right )}^2}{d+e\,x} \,d x \]
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