Optimal. Leaf size=186 \[ \frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5785, 5783,
5780, 5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5780
Rule 5783
Rule 5785
Rubi steps
\begin {align*} \int \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)} \, dx &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {1+a^2 x^2}}-\frac {\left (a \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{4 \sqrt {1+a^2 x^2}}\\ &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a \sqrt {1+a^2 x^2}}\\ &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a \sqrt {1+a^2 x^2}}\\ &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt {1+a^2 x^2}}\\ &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}+\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}\\ &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}+\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}\\ &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 104, normalized size = 0.56 \begin {gather*} \frac {\sqrt {c \left (1+a^2 x^2\right )} \left (16 \sinh ^{-1}(a x)^2-3 \sqrt {2} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \sinh ^{-1}(a x)\right )-3 \sqrt {2} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 \sinh ^{-1}(a x)\right )\right )}{48 a \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {a^{2} c \,x^{2}+c}\, \sqrt {\arcsinh \left (a x \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \sqrt {c \left (a^{2} x^{2} + 1\right )} \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \sqrt {\mathrm {asinh}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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