Integrand size = 23, antiderivative size = 385 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \]
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Time = 0.39 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5859, 12, 5776, 5809, 5816, 4267, 2611, 6744, 2320, 6724, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {8 b^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^4}+\frac {4 b \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3}{3 d e^4}-\frac {4 b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^4}+\frac {4 b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^4}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e^4}-\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {4 b^4 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \]
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Rule 12
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4267
Rule 5776
Rule 5809
Rule 5816
Rule 5859
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{e^4 x^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{x^4} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}+\frac {(4 b) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x^3 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e^4} \\ & = -\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {(2 b) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e^4}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x^2} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {(2 b) \text {Subst}\left (\int (a+b x)^3 \text {csch}(x) \, dx,x,\text {arcsinh}(c+d x)\right )}{3 d e^4}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}+\frac {\left (4 b^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c+d x))}{d e^4} \\ & = -\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}+\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}-\frac {\left (4 b^4\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}+\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {\left (4 b^4\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}-\frac {\left (4 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {\left (4 b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1198\) vs. \(2(385)=770\).
Time = 8.24 (sec) , antiderivative size = 1198, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {a^4}{3 d e^4 (c+d x)^3}+\frac {a^2 b^2 \left (-8 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-\frac {2 \left (-2+4 \text {arcsinh}(c+d x)^2+2 \cosh (2 \text {arcsinh}(c+d x))-3 (c+d x) \text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )+3 (c+d x) \text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )-4 (c+d x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+2 \text {arcsinh}(c+d x) \sinh (2 \text {arcsinh}(c+d x))+\text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right ) \sinh (3 \text {arcsinh}(c+d x))-\text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right ) \sinh (3 \text {arcsinh}(c+d x))\right )}{(c+d x)^3}\right )}{4 d e^4}+\frac {a b^3 \left (-24 \text {arcsinh}(c+d x) \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )+4 \text {arcsinh}(c+d x)^3 \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-6 \text {arcsinh}(c+d x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-(c+d x) \text {arcsinh}(c+d x)^3 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-24 \text {arcsinh}(c+d x)^2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+48 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )+48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c+d x)}\right )-6 \text {arcsinh}(c+d x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {16 \text {arcsinh}(c+d x)^3 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )}{(c+d x)^3}+24 \text {arcsinh}(c+d x) \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-4 \text {arcsinh}(c+d x)^3 \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )}{12 d e^4}+\frac {b^4 \left (-2 \pi ^4+4 \text {arcsinh}(c+d x)^4-24 \text {arcsinh}(c+d x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )+2 \text {arcsinh}(c+d x)^4 \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-4 \text {arcsinh}(c+d x)^3 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {1}{2} (c+d x) \text {arcsinh}(c+d x)^4 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )+96 \text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-96 \text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+16 \text {arcsinh}(c+d x)^3 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )-16 \text {arcsinh}(c+d x)^3 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )-48 \left (-2+\text {arcsinh}(c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-96 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )-48 \text {arcsinh}(c+d x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )-96 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )+96 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )-96 \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(c+d x)}\right )-96 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )-4 \text {arcsinh}(c+d x)^3 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {8 \text {arcsinh}(c+d x)^4 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )}{(c+d x)^3}+24 \text {arcsinh}(c+d x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-2 \text {arcsinh}(c+d x)^4 \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )}{24 d e^4}+\frac {4 a^3 b \left (\frac {1}{12} \text {arcsinh}(c+d x) \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {1}{24} \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {1}{24} \text {arcsinh}(c+d x) \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right ) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )+\frac {1}{6} \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-\frac {1}{6} \log \left (\sinh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-\frac {1}{24} \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {1}{12} \text {arcsinh}(c+d x) \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {1}{24} \text {arcsinh}(c+d x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right ) \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )}{d e^4} \]
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Time = 0.56 (sec) , antiderivative size = 883, normalized size of antiderivative = 2.29
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(883\) |
default | \(\text {Expression too large to display}\) | \(883\) |
parts | \(\text {Expression too large to display}\) | \(894\) |
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{4}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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