Optimal. Leaf size=152 \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{a \sqrt {\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5696, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{a \sqrt {\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5669
Rule 5696
Rubi steps
\begin {align*} \int \frac {\sqrt {c+a^2 c x^2}}{\sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (4 a \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{\sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (4 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (4 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (2 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\sqrt {c+a^2 c x^2} \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}+\frac {\sqrt {c+a^2 c x^2} \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\left (2 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a \sqrt {1+a^2 x^2}}+\frac {\left (2 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{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 )}{a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 115, normalized size = 0.76 \[ -\frac {\sqrt {a^2 c x^2+c} \left (4 a^2 x^2+\sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )-\sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+4\right )}{2 a \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a^{2} c \,x^{2}+c}}{\arcsinh \left (a x \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,a^2\,x^2+c}}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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