Integrand size = 12, antiderivative size = 116 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c} \]
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Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5773, 5819, 3389, 2211, 2236, 2235} \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5773
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(2 c) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c}+\frac {\text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c}+\frac {2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {a+b \text {arcsinh}(c x)}{b}} \left (-e^{a/b} \left (1+e^{2 \text {arcsinh}(c x)}\right )+e^{\frac {2 a}{b}+\text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+e^{\text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int \frac {1}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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