Optimal. Leaf size=92 \[ \frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d} \]
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Rubi [A] time = 0.128954, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5863, 5657, 3307, 2180, 2205, 2204} \[ \frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 5863
Rule 5657
Rule 3307
Rule 2180
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{2 b d}\\ &=\frac{\operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d}+\frac{\operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d}\\ &=\frac{e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}+\frac{e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.107971, size = 111, normalized size = 1.21 \[ \frac{e^{-\frac{a}{b}} \left (\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )-e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )}{2 d \sqrt{a+b \sinh ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a+b{\it Arcsinh} \left ( dx+c \right ) }}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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