Integrand size = 27, antiderivative size = 204 \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6} \]
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Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5818, 5780, 5556, 3384, 3379, 3382} \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {x^5}{b c (a+b \text {arcsinh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5780
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \int \frac {x^4}{a+b \text {arcsinh}(c x)} \, dx}{b c} \\ & = -\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^6} \\ & = -\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^6} \\ & = -\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^6}+\frac {5 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^6}-\frac {15 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^6} \\ & = -\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {\left (5 \cosh \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 x)\right )}{8 b^2 c^6}-\frac {\left (15 \cosh \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 x)\right )}{16 b^2 c^6}+\frac {\left (5 \cosh \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 x)\right )}{16 b^2 c^6}-\frac {\left (5 \sinh \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 x)\right )}{8 b^2 c^6}+\frac {\left (15 \sinh \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 x)\right )}{16 b^2 c^6}-\frac {\left (5 \sinh \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 x)\right )}{16 b^2 c^6} \\ & = -\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.77 \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \left (2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )\right )}{16 b^2 c^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(632\) vs. \(2(192)=384\).
Time = 0.27 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.10
method | result | size |
default | \(-\frac {16 c^{5} x^{5}-16 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+20 c^{3} x^{3}-12 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+5 c x -\sqrt {c^{2} x^{2}+1}}{32 c^{6} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )}{32 c^{6} b^{2}}+\frac {\frac {5 c^{3} x^{3}}{8}-\frac {5 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{8}+\frac {15 c x}{32}-\frac {5 \sqrt {c^{2} x^{2}+1}}{32}}{c^{6} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {15 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{32 c^{6} b^{2}}-\frac {5 \left (-\sqrt {c^{2} x^{2}+1}+c x \right )}{16 c^{6} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{16 c^{6} b^{2}}-\frac {5 \left (\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a}{b}} b +\operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a}{b}} a +b c x +\sqrt {c^{2} x^{2}+1}\, b \right )}{16 c^{6} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {\frac {5 b \,c^{3} x^{3}}{8}+\frac {5 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}}{8}+\frac {15 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} b}{32}+\frac {15 \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} a}{32}+\frac {15 b c x}{32}+\frac {5 \sqrt {c^{2} x^{2}+1}\, b}{32}}{c^{6} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {16 b \,c^{5} x^{5}+16 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+20 b \,c^{3} x^{3}+12 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+5 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {5 a}{b}} b +5 \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {5 a}{b}} a +5 b c x +\sqrt {c^{2} x^{2}+1}\, b}{32 c^{6} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\) | \(633\) |
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\[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{5}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{5}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{5}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^5}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]
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