Integrand size = 14, antiderivative size = 122 \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=-\frac {2 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \]
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Time = 0.17 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5773, 5819, 3389, 2211, 2236, 2235} \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 \sqrt {(c+d x)^2+1}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5773
Rule 5819
Rule 5858
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {2 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d 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+d x)\right )}{b^2 d} \\ & = -\frac {2 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d 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+d x)\right )}{b^2 d}+\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+d x)\right )}{b^2 d} \\ & = -\frac {2 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2 d}+\frac {2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2 d} \\ & = -\frac {2 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {e^{-\frac {a+b \text {arcsinh}(c+d x)}{b}} \left (-e^{a/b} \left (1+e^{2 \text {arcsinh}(c+d x)}\right )+e^{\frac {2 a}{b}+\text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+e^{\text {arcsinh}(c+d x)} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{b d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
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\[\int \frac {1}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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