Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^4 \operatorname {PolyLog}\left (5,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]
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Time = 0.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5775, 3797, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=-\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e}-\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e}-\frac {2 b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3}{d e}+\frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {\log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^4}{d e}-\frac {3 b^4 \operatorname {PolyLog}\left (5,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
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 x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{x} \, dx,x,c+d x\right )}{d e} \\ & = -\frac {\text {Subst}\left (\int x^4 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {2 \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^4}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {4 \text {Subst}\left (\int x^3 \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}+\frac {(6 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^4 \operatorname {PolyLog}\left (5,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c+d x))^5}{5 b}+(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )+2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )-3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c+d x)}\right )+3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(c+d x)}\right )-\frac {3}{2} b^4 \operatorname {PolyLog}\left (5,e^{2 \text {arcsinh}(c+d x)}\right )}{d e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(860\) vs. \(2(222)=444\).
Time = 0.54 (sec) , antiderivative size = 861, normalized size of antiderivative = 4.63
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(861\) |
default | \(\text {Expression too large to display}\) | \(861\) |
parts | \(\text {Expression too large to display}\) | \(872\) |
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\frac {\int \frac {a^{4}}{c + d x}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{c\,e+d\,e\,x} \,d x \]
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