Integrand size = 21, antiderivative size = 515 \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2 \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.37 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5788, 5787, 5797, 3799, 2221, 2611, 2320, 6724, 5798, 266, 267} \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {8 \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {a^2 c x^2+c}}+\frac {4 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {a^2 c x^2+c}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {a^2 c x^2+c}}+\frac {8 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {a^2 c x^2+c}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c}}-\frac {8 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2 \log \left (e^{2 \text {arcsinh}(a x)}+1\right )}{5 a c^3 \sqrt {a^2 c x^2+c}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (a^2 c x^2+c\right )^{5/2}}-\frac {1}{20 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^3 \sqrt {a^2 c x^2+c}} \]
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Rule 266
Rule 267
Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5787
Rule 5788
Rule 5797
Rule 5798
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \text {arcsinh}(a x)^2}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \text {arcsinh}(a x)^2}{\left (1+a^2 x^2\right )^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {\text {arcsinh}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2} \, dx}{10 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \text {arcsinh}(a x)^2}{1+a^2 x^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \tanh (x) \, dx,x,\text {arcsinh}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2 \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2 \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2 \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \text {arcsinh}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \text {arcsinh}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \text {arcsinh}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2 \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.58 \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {-\frac {3}{\sqrt {1+a^2 x^2}}-60 a x \text {arcsinh}(a x)-\frac {6 a x \text {arcsinh}(a x)}{1+a^2 x^2}+\frac {9 \text {arcsinh}(a x)^2}{\left (1+a^2 x^2\right )^{3/2}}+\frac {24 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}}+32 a x \text {arcsinh}(a x)^3+\frac {12 a x \text {arcsinh}(a x)^3}{\left (1+a^2 x^2\right )^2}+\frac {16 a x \text {arcsinh}(a x)^3}{1+a^2 x^2}-32 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3-96 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2 \log \left (1+e^{-2 \text {arcsinh}(a x)}\right )+30 \sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )+96 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(a x)}\right )+48 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(a x)}\right )}{60 a c^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 888, normalized size of antiderivative = 1.72
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+20 a^{3} x^{3}-16 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+15 a x -8 \sqrt {a^{2} x^{2}+1}\right ) \left (24-1590 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )-1368 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}-1410 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )+105 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}+160 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3}+45 a x \sqrt {a^{2} x^{2}+1}-744 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}-1020 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-495 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +256 \operatorname {arcsinh}\left (a x \right )^{3}-480 \,\operatorname {arcsinh}\left (a x \right )-264 \operatorname {arcsinh}\left (a x \right )^{2}+96 a^{2} x^{2}-192 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}-192 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}-840 \operatorname {arcsinh}\left (a x \right )^{2} a^{6} x^{6}+84 x^{5} a^{5} \sqrt {a^{2} x^{2}+1}-984 \operatorname {arcsinh}\left (a x \right )^{2} a^{2} x^{2}+144 a^{4} x^{4}-936 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}-372 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -756 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}+96 a^{6} x^{6}+24 \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}+24 a^{8} x^{8}-192 \,\operatorname {arcsinh}\left (a x \right ) a^{8} x^{8}-852 \,\operatorname {arcsinh}\left (a x \right ) a^{6} x^{6}-192 \operatorname {arcsinh}\left (a x \right )^{2} a^{8} x^{8}+380 \operatorname {arcsinh}\left (a x \right )^{3} a^{2} x^{2}\right )}{60 \left (40 a^{10} x^{10}+215 a^{8} x^{8}+469 a^{6} x^{6}+517 a^{4} x^{4}+287 a^{2} x^{2}+64\right ) a \,c^{4}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (a x +\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {16 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{3}}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{5 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{5 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {4 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{5 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}\) | \(888\) |
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\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{7/2}} \,d x \]
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