Optimal. Leaf size=237 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {16 \sqrt {a x-1} \sqrt {a x+1}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}-\frac {24 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.85, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5656, 5781} \[ \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {a x-1} \sqrt {a x+1}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}-\frac {24 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5656
Rule 5666
Rule 5668
Rule 5775
Rule 5781
Rubi steps
\begin {align*} \int \frac {x^2}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac {4 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (6 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac {12}{5} \int \frac {x^2}{\cosh ^{-1}(a x)^{3/2}} \, dx-\frac {8 \int \frac {1}{\cosh ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {24 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \sqrt {\cosh ^{-1}(a x)}}-\frac {24 \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}-\frac {3 \cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}-\frac {16 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx}{15 a}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {24 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac {18 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {24 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \sqrt {\cosh ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^3}-\frac {8 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {24 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}-\frac {16 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {6 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {6 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {18 \operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {18 \operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {8 x}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {4 x^3}{5 \cosh ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{15 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {24 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \sqrt {\cosh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^3}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 286, normalized size = 1.21 \[ \frac {e^{-3 \cosh ^{-1}(a x)} \left (-e^{2 \cosh ^{-1}(a x)} \left (2 e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-2 \cosh ^{-1}(a x)^2+3 \sqrt {\frac {a x-1}{a x+1}} (a x+1) e^{\cosh ^{-1}(a x)}+e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)-2 e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )\right )-3 \left (6 e^{6 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-6 \cosh ^{-1}(a x)^2+e^{6 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)+e^{3 \cosh ^{-1}(a x)} \sinh \left (3 \cosh ^{-1}(a x)\right )-6 \sqrt {3} e^{3 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-3 \cosh ^{-1}(a x)\right )+6 \sqrt {3} e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \cosh ^{-1}(a x)\right )\right )\right )}{30 a^3 \cosh ^{-1}(a x)^{5/2}} \]
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 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {x^{2}}{\mathrm {arccosh}\left (a x \right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {acosh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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