Optimal. Leaf size=139 \[ -\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.40, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5664
Rule 5781
Rubi steps
\begin {align*} \int x^3 \sqrt {\cosh ^{-1}(a x)} \, dx &=\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {1}{8} a \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx\\ &=\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^4}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a^4}-\frac {\operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a^4}-\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^4}-\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{64 a^4}-\frac {\operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{64 a^4}-\frac {\operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{16 a^4}-\frac {\operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{16 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 101, normalized size = 0.73 \[ \frac {\sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-4 \cosh ^{-1}(a x)\right )+4 \sqrt {2} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt {-\cosh ^{-1}(a x)} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},2 \cosh ^{-1}(a x)\right )+\Gamma \left (\frac {3}{2},4 \cosh ^{-1}(a x)\right )\right )}{128 a^4 \sqrt {-\cosh ^{-1}(a x)}} \]
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(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {\mathrm {arccosh}\left (a x \right )}\, dx \]
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 \[ \int x^{3} \sqrt {\operatorname {arcosh}\left (a x\right )}\,{d x} \]
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 \[ \int x^3\,\sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \]
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 \[ \int x^{3} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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