Optimal. Leaf size=93 \[ \frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}+\frac {3 \sinh \left (a+\frac {b}{x^2}\right )}{4 b^2 x}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5346, 5324, 5325, 5298, 2204, 2205} \[ \frac {3 \sqrt {\pi } e^{-a} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\pi } e^a \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}+\frac {3 \sinh \left (a+\frac {b}{x^2}\right )}{4 b^2 x}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5324
Rule 5325
Rule 5346
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx &=-\operatorname {Subst}\left (\int x^4 \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}+\frac {3 \operatorname {Subst}\left (\int x^2 \cosh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}+\frac {3 \sinh \left (a+\frac {b}{x^2}\right )}{4 b^2 x}-\frac {3 \operatorname {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )}{4 b^2}\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}+\frac {3 \sinh \left (a+\frac {b}{x^2}\right )}{4 b^2 x}+\frac {3 \operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )}{8 b^2}-\frac {3 \operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )}{8 b^2}\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}+\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}-\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}+\frac {3 \sinh \left (a+\frac {b}{x^2}\right )}{4 b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 97, normalized size = 1.04 \[ \frac {3 \sqrt {\pi } x^3 (\cosh (a)-\sinh (a)) \text {erf}\left (\frac {\sqrt {b}}{x}\right )-3 \sqrt {\pi } x^3 (\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+4 \sqrt {b} \left (3 x^2 \sinh \left (a+\frac {b}{x^2}\right )-2 b \cosh \left (a+\frac {b}{x^2}\right )\right )}{16 b^{5/2} x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 313, normalized size = 3.37 \[ -\frac {6 \, b x^{2} - 2 \, {\left (3 \, b x^{2} - 2 \, b^{2}\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 3 \, \sqrt {\pi } {\left (x^{3} \cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x^{3} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (x^{3} \cosh \relax (a) + x^{3} \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - 3 \, \sqrt {\pi } {\left (x^{3} \cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - x^{3} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (x^{3} \cosh \relax (a) - x^{3} \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) - 4 \, {\left (3 \, b x^{2} - 2 \, b^{2}\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - 2 \, {\left (3 \, b x^{2} - 2 \, b^{2}\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + 4 \, b^{2}}{16 \, {\left (b^{3} x^{3} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + b^{3} x^{3} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]
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 \[ \int \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 117, normalized size = 1.26 \[ -\frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}}}{4 b \,x^{3}}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}}}{8 b^{2} x}+\frac {3 \,{\mathrm e}^{-a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {b}}{x}\right )}{16 b^{\frac {5}{2}}}-\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}}}{4 x^{3} b}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}}}{8 b^{2} x}-\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{16 b^{2} \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 62, normalized size = 0.67 \[ -\frac {1}{10} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (\frac {7}{2}, \frac {b}{x^{2}}\right )}{x^{7} \left (\frac {b}{x^{2}}\right )^{\frac {7}{2}}} + \frac {e^{a} \Gamma \left (\frac {7}{2}, -\frac {b}{x^{2}}\right )}{x^{7} \left (-\frac {b}{x^{2}}\right )^{\frac {7}{2}}}\right )} - \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{5 \, x^{5}} \]
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 \[ \int \frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{x^6} \,d x \]
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 \[ \int \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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