Integrand size = 27, antiderivative size = 277 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}-\frac {9 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4} \]
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Time = 0.58 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5814, 5819, 5556, 3384, 3379, 3382} \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}-\frac {9 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5814
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}+\frac {3 \int \frac {x^2 \left (1+c^2 x^2\right )}{a+b \text {arcsinh}(c x)} \, dx}{b c}+\frac {(7 c) \int \frac {x^4 \left (1+c^2 x^2\right )}{a+b \text {arcsinh}(c x)} \, dx}{b} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^4}+\frac {7 \text {Subst}\left (\int \frac {\cosh ^3\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^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}+\frac {\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^4}+\frac {7 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}-\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}+\frac {3 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}+\frac {7 \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {7 \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 )}{64 b^2 c^4}+\frac {3 \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^4}+\frac {3 \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^4}-\frac {21 \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 )}{64 b^2 c^4}+\frac {21 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {3 \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^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}+\frac {\left (21 \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 )}{64 b^2 c^4}-\frac {\left (3 \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^4}+\frac {\left (3 \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^4}-\frac {\left (21 \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 )}{64 b^2 c^4}-\frac {\left (7 \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 )}{64 b^2 c^4}+\frac {\left (3 \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^4}+\frac {\left (7 \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {\left (21 \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 )}{64 b^2 c^4}+\frac {\left (3 \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^4}-\frac {\left (3 \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^4}+\frac {\left (21 \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 )}{64 b^2 c^4}+\frac {\left (7 \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 )}{64 b^2 c^4}-\frac {\left (3 \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^4}-\frac {\left (7 \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}-\frac {9 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.44 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {-64 b c^3 x^3-128 b c^5 x^5-64 b c^7 x^7-3 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-9 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 a \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 b \text {arcsinh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+3 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+9 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 a \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 b \text {arcsinh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b^2 c^4 (a+b \text {arcsinh}(c x))} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(261)=522\).
Time = 0.31 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.46
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\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{3} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^3\,{\left (c^2\,x^2+1\right )}^{3/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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