Optimal. Leaf size=302 \[ \frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{16 a \sqrt {a x-1} \sqrt {a x+1}} \]
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Rubi [A] time = 0.62, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ \frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{16 a \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5664
Rule 5676
Rule 5683
Rule 5713
Rule 5781
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\cosh ^{-1}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \sqrt {\cosh ^{-1}(a x)} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 136, normalized size = 0.45 \[ \frac {\sqrt {c-a^2 c x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-8 \sqrt {\cosh ^{-1}(a x)} \left (16 \cosh ^{-1}(a x)^2+15 \cosh \left (2 \cosh ^{-1}(a x)\right )-20 \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{640 a \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
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(-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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maple [F] time = 0.70, size = 0, normalized size = 0.00 \[ \int \sqrt {-a^{2} c \,x^{2}+c}\, \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {acosh}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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 \[ \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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