Optimal. Leaf size=47 \[ -\frac {\cosh \left (a+\frac {b}{x^2}\right )}{b^3}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^4}+\frac {\sinh \left (a+\frac {b}{x^2}\right )}{b^2 x^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5428, 3377,
2718} \begin {gather*} -\frac {\cosh \left (a+\frac {b}{x^2}\right )}{b^3}+\frac {\sinh \left (a+\frac {b}{x^2}\right )}{b^2 x^2}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5428
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^7} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^4}+\frac {\text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\frac {1}{x^2}\right )}{b}\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^4}+\frac {\sinh \left (a+\frac {b}{x^2}\right )}{b^2 x^2}-\frac {\text {Subst}\left (\int \sinh (a+b x) \, dx,x,\frac {1}{x^2}\right )}{b^2}\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{b^3}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^4}+\frac {\sinh \left (a+\frac {b}{x^2}\right )}{b^2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.94 \begin {gather*} \frac {-\left (\left (b^2+2 x^4\right ) \cosh \left (a+\frac {b}{x^2}\right )\right )+2 b x^2 \sinh \left (a+\frac {b}{x^2}\right )}{2 b^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 73, normalized size = 1.55
method | result | size |
risch | \(-\frac {\left (2 x^{4}-2 x^{2} b +b^{2}\right ) {\mathrm e}^{\frac {a \,x^{2}+b}{x^{2}}}}{4 b^{3} x^{4}}-\frac {\left (2 x^{4}+2 x^{2} b +b^{2}\right ) {\mathrm e}^{-\frac {a \,x^{2}+b}{x^{2}}}}{4 b^{3} x^{4}}\) | \(73\) |
meijerg | \(-\frac {2 \sqrt {\pi }\, \cosh \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {b^{2}}{2 x^{4}}+1\right ) \cosh \left (\frac {b}{x^{2}}\right )}{2 \sqrt {\pi }}-\frac {b \sinh \left (\frac {b}{x^{2}}\right )}{2 \sqrt {\pi }\, x^{2}}\right )}{b^{3}}-\frac {2 i \sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {i b \cosh \left (\frac {b}{x^{2}}\right )}{2 \sqrt {\pi }\, x^{2}}-\frac {i \left (\frac {3 b^{2}}{2 x^{4}}+3\right ) \sinh \left (\frac {b}{x^{2}}\right )}{6 \sqrt {\pi }}\right )}{b^{3}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.30, size = 47, normalized size = 1.00 \begin {gather*} -\frac {1}{12} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (4, \frac {b}{x^{2}}\right )}{b^{4}} + \frac {e^{a} \Gamma \left (4, -\frac {b}{x^{2}}\right )}{b^{4}}\right )} - \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 50, normalized size = 1.06 \begin {gather*} \frac {2 \, b x^{2} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - {\left (2 \, x^{4} + b^{2}\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.56, size = 51, normalized size = 1.09 \begin {gather*} \begin {cases} - \frac {\cosh {\left (a + \frac {b}{x^{2}} \right )}}{2 b x^{4}} + \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{b^{2} x^{2}} - \frac {\cosh {\left (a + \frac {b}{x^{2}} \right )}}{b^{3}} & \text {for}\: b \neq 0 \\- \frac {\sinh {\left (a \right )}}{6 x^{6}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.44, size = 74, normalized size = 1.57 \begin {gather*} -\frac {{\mathrm {e}}^{a+\frac {b}{x^2}}\,\left (\frac {1}{4\,b}-\frac {x^2}{2\,b^2}+\frac {x^4}{2\,b^3}\right )}{x^4}-\frac {{\mathrm {e}}^{-a-\frac {b}{x^2}}\,\left (\frac {1}{4\,b}+\frac {x^2}{2\,b^2}+\frac {x^4}{2\,b^3}\right )}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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