Integrand size = 23, antiderivative size = 256 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 d}+\frac {9 e^4 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 d}-\frac {5 e^4 \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d}+\frac {5 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d} \]
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Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5859, 12, 5778, 3384, 3379, 3382} \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d}+\frac {5 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d}-\frac {e^4 (c+d x)^4 \sqrt {(c+d x)^2+1}}{b d (a+b \text {arcsinh}(c+d x))} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5778
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^4 x^4}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {e^4 \text {Subst}\left (\int \left (-\frac {5 \sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}+\frac {9 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}-\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {e^4 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^2 d}-\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b^2 d}+\frac {\left (9 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {\left (e^4 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b^2 d}+\frac {\left (5 e^4 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b^2 d}-\frac {\left (e^4 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b^2 d}-\frac {\left (5 e^4 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 d}+\frac {9 e^4 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 d}-\frac {5 e^4 \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d}+\frac {5 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 b^2 d} \\ \end{align*}
Time = 1.25 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.10 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\frac {e^4 \left (-\frac {16 b (c+d x)^4 \sqrt {1+(c+d x)^2}}{a+b \text {arcsinh}(c+d x)}+16 \left (3 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )-5 \left (10 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-5 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-10 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{16 b^2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(242)=484\).
Time = 0.97 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.35
method | result | size |
derivativedivides | \(\frac {\frac {\left (-16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+16 \left (d x +c \right )^{5}-12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+20 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {5 e^{4} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {5 a}{b}\right )}{32 b^{2}}-\frac {3 \left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{4} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{4}}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {e^{4} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{2}}+\frac {3 e^{4} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {9 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {e^{4} \left (16 \left (d x +c \right )^{5}+20 \left (d x +c \right )^{3}+16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c +12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {5 e^{4} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{d}\) | \(602\) |
default | \(\frac {\frac {\left (-16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+16 \left (d x +c \right )^{5}-12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+20 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {5 e^{4} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {5 a}{b}\right )}{32 b^{2}}-\frac {3 \left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{4} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{4}}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {e^{4} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{2}}+\frac {3 e^{4} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {9 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {e^{4} \left (16 \left (d x +c \right )^{5}+20 \left (d x +c \right )^{3}+16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c +12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {5 e^{4} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{d}\) | \(602\) |
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\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2} \,d x \]
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