Optimal. Leaf size=157 \[ \frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
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Rubi [A] time = 0.50, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5676} \[ \frac {8 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5666
Rule 5668
Rule 5676
Rule 5775
Rubi steps
\begin {align*} \int \frac {x}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac {2 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (4 a) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16}{15} \int \frac {x}{\cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {32 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}\\ &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}+\frac {16 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}\\ &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {32 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac {32 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}\\ &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^2}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 91, normalized size = 0.58 \[ -\frac {-8 \sqrt {2 \pi } \left (\text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )\right )+\frac {4 \cosh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{3/2}}+\frac {\left (16 \cosh ^{-1}(a x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{5/2}}}{15 a^2} \]
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 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 153, normalized size = 0.97 \[ -\frac {\sqrt {2}\, \left (16 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, x a +4 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x^{2} a^{2}+3 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, x a -2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }-8 \mathrm {arccosh}\left (a x \right )^{3} \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-8 \mathrm {arccosh}\left (a x \right )^{3} \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{15 \sqrt {\pi }\, a^{2} \mathrm {arccosh}\left (a x \right )^{3}} \]
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 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {x}{\operatorname {acosh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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