Optimal. Leaf size=170 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {\cosh ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.10, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5904, 5887,
5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\cosh ^{-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 5887
Rule 5904
Rubi steps
\begin {align*} \int \frac {\sqrt {c-a^2 c x^2}}{\cosh ^{-1}(a x)^{3/2}} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {-1+a x} \sqrt {1+a x}}{\cosh ^{-1}(a x)^{3/2}} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (4 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}-\frac {\sqrt {c-a^2 c x^2} \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}-\frac {\left (2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 127, normalized size = 0.75 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (4-4 a^2 x^2-\sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+\sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )\right )}{2 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\cosh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}\, dx\]
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 \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 \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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