Optimal. Leaf size=394 \[ -\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac {4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{10 a} \]
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Rubi [A] time = 1.82, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5664, 5759, 5718, 5654, 5781, 3307, 2180, 2204, 2205, 3312} \[ -\frac {15 \sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^3}+\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}-\frac {4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{15 a^5}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{10 a} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5654
Rule 5664
Rule 5718
Rule 5759
Rule 5781
Rubi steps
\begin {align*} \int x^4 \cosh ^{-1}(a x)^{5/2} \, dx &=\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {1}{2} a \int \frac {x^5 \cosh ^{-1}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}+\frac {3}{20} \int x^4 \sqrt {\cosh ^{-1}(a x)} \, dx-\frac {2 \int \frac {x^3 \cosh ^{-1}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{5 a}\\ &=\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {4 \int \frac {x \cosh ^{-1}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{15 a^3}+\frac {\int x^2 \sqrt {\cosh ^{-1}(a x)} \, dx}{5 a^2}-\frac {1}{200} (3 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx\\ &=\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cosh ^5(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{200 a^5}+\frac {2 \int \sqrt {\cosh ^{-1}(a x)} \, dx}{5 a^4}-\frac {\int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx}{30 a}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {3 \operatorname {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 \sqrt {x}}+\frac {5 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{200 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\cosh ^3(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{30 a^5}-\frac {\int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx}{5 a^3}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3200 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{640 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{320 a^5}-\frac {\operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{30 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^5}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6400 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6400 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1280 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1280 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{640 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{640 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{120 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{40 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{10 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {3 \operatorname {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{3200 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{240 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{240 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{320 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{80 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{80 a^5}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^5}-\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^5}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {67 \sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac {67 \sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{640 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac {\operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{120 a^5}-\frac {\operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{120 a^5}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{40 a^5}-\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{40 a^5}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{5 a^4}+\frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\cosh ^{-1}(a x)}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{240 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\cosh ^{-1}(a x)}\right )}{6400 a^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 162, normalized size = 0.41 \[ \frac {27 \sqrt {5} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-5 \cosh ^{-1}(a x)\right )+625 \sqrt {3} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-3 \cosh ^{-1}(a x)\right )+33750 \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-\cosh ^{-1}(a x)\right )+33750 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},\cosh ^{-1}(a x)\right )+625 \sqrt {3} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},3 \cosh ^{-1}(a x)\right )+27 \sqrt {5} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},5 \cosh ^{-1}(a x)\right )}{540000 a^5 \sqrt {-\cosh ^{-1}(a x)}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int x^{4} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \int x^{4} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int x^4\,{\mathrm {acosh}\left (a\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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