Optimal. Leaf size=51 \[ -\frac {1}{4 x^2}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3460, 3391, 30}
\begin {gather*} -\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x}-\frac {1}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3391
Rule 3460
Rubi steps
\begin {align*} \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\text {Subst}\left (\int x \sin ^2(a+b x) \, dx,x,\frac {1}{x}\right )\\ &=\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}-\frac {1}{2} \text {Subst}\left (\int x \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4 x^2}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 0.84 \begin {gather*} \frac {x^2 \cos \left (2 \left (a+\frac {b}{x}\right )\right )-2 b \left (b-x \sin \left (2 \left (a+\frac {b}{x}\right )\right )\right )}{8 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs.
\(2(45)=90\).
time = 0.05, size = 97, normalized size = 1.90
method | result | size |
risch | \(-\frac {1}{4 x^{2}}+\frac {\cos \left (\frac {2 a x +2 b}{x}\right )}{8 b^{2}}+\frac {\sin \left (\frac {2 a x +2 b}{x}\right )}{4 b x}\) | \(42\) |
derivativedivides | \(-\frac {\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{4}-a \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )}{b^{2}}\) | \(97\) |
default | \(-\frac {\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{4}-a \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )}{b^{2}}\) | \(97\) |
norman | \(\frac {-\frac {1}{4}+\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b}-\frac {x^{2} \left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{b^{2}}-\frac {\left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{4}-\frac {x \left (\tan ^{3}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )^{2} x^{2}}\) | \(110\) |
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 = 0.37, size = 68, normalized size = 1.33 \begin {gather*} \frac {{\left ({\left (\Gamma \left (2, \frac {2 i \, b}{x}\right ) + \Gamma \left (2, -\frac {2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) - {\left (i \, \Gamma \left (2, \frac {2 i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac {2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right )\right )} x^{2} - 4 \, b^{2}}{16 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 60, normalized size = 1.18 \begin {gather*} \frac {2 \, x^{2} \cos \left (\frac {a x + b}{x}\right )^{2} + 4 \, b x \cos \left (\frac {a x + b}{x}\right ) \sin \left (\frac {a x + b}{x}\right ) - 2 \, b^{2} - x^{2}}{8 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 391 vs.
\(2 (37) = 74\).
time = 1.61, size = 391, normalized size = 7.67 \begin {gather*} \begin {cases} - \frac {b^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {2 b^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {b^{2}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {4 b x \tan ^{3}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} + \frac {4 b x \tan {\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {4 x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} & \text {for}\: b \neq 0 \\- \frac {\sin ^{2}{\left (a \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.72, size = 77, normalized size = 1.51 \begin {gather*} -\frac {2 \, a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {4 \, {\left (a x + b\right )} a}{x} - \frac {2 \, {\left (a x + b\right )} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {2 \, {\left (a x + b\right )}^{2}}{x^{2}} - \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{8 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.62, size = 41, normalized size = 0.80 \begin {gather*} \frac {\cos \left (2\,a+\frac {2\,b}{x}\right )}{8\,b^2}-\frac {1}{4\,x^2}+\frac {\sin \left (2\,a+\frac {2\,b}{x}\right )}{4\,b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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