Optimal. Leaf size=145 \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}+\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)} \]
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Rubi [A] time = 0.43, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5663, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}+\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5663
Rule 5779
Rubi steps
\begin {align*} \int x \sqrt {a+b \sinh ^{-1}(c x)} \, dx &=\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx\\ &=\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2}\\ &=\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2}\\ &=\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2}\\ &=\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2}-\frac {b \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2}\\ &=\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\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 )}{8 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 )}{8 c^2}\\ &=\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {\sqrt {b} 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 )}{16 c^2}-\frac {\sqrt {b} 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 )}{16 c^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 127, normalized size = 0.88 \[ \frac {e^{-\frac {2 a}{b}} \sqrt {a+b \sinh ^{-1}(c x)} \left (\sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{8 \sqrt {2} c^2 \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}} \]
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 \[ \int \sqrt {b \operatorname {arsinh}\left (c x\right ) + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int x \sqrt {a +b \arcsinh \left (c x \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {arsinh}\left (c x\right ) + a} x\,{d x} \]
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 \[ \int x\,\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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 \[ \int x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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