Optimal. Leaf size=228 \[ -\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.86, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {Erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {Erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5668
Rule 5670
Rule 5775
Rubi steps
\begin {align*} \int \frac {x^4}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac {8 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {100}{3} \int \frac {x^4}{\sqrt {\cosh ^{-1}(a x)}} \, dx-\frac {16 \int \frac {x^2}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {100 \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {x}}+\frac {\sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {100 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{8 \sqrt {x}}+\frac {3 \sinh (3 x)}{16 \sqrt {x}}+\frac {\sinh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}+\frac {25 \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac {4 \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {4 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {25 \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {2 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac {25 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac {25 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {25 \operatorname {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {25 \operatorname {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {4 \operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}+\frac {4 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}-\frac {4 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}-\frac {4 \operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a^5}-\frac {25 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^5}+\frac {25 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^5}-\frac {25 \operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^5}+\frac {25 \operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^5}\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\cosh ^{-1}(a x)}}-\frac {20 x^5}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{24 a^5}\\ \end {align*}
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Mathematica [A] time = 1.66, size = 278, normalized size = 1.22 \[ -\frac {2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)+2 e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\sinh \left (5 \cosh ^{-1}(a x)\right )+2 \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )+5 \cosh ^{-1}(a x) \left (e^{-5 \cosh ^{-1}(a x)}+e^{5 \cosh ^{-1}(a x)}-\sqrt {5} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-5 \cosh ^{-1}(a x)\right )-\sqrt {5} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},5 \cosh ^{-1}(a x)\right )\right )+3 \left (3 e^{-3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+3 e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\sinh \left (3 \cosh ^{-1}(a x)\right )+3 \sqrt {3} \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-3 \cosh ^{-1}(a x)\right )-3 \sqrt {3} \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \cosh ^{-1}(a x)\right )\right )}{24 a^5 \cosh ^{-1}(a x)^{3/2}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^{4}}{\mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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