Optimal. Leaf size=107 \[ \frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}-\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]
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Rubi [A] time = 0.18, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5669, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}-\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5669
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2}+\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2}\\ &=-\frac {\operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 b c^2}+\frac {\operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 b c^2}\\ &=-\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 108, normalized size = 1.01 \[ \frac {e^{-\frac {2 a}{b}} \left (\sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{4 \sqrt {2} c^2 \sqrt {a+b \sinh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a +b \arcsinh \left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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