Optimal. Leaf size=87 \[ -\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{4 b^3}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{2 b^2 x}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^2}+\frac {1}{4 b^2 x}-\frac {1}{6 x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3379, 3311, 30, 2635, 8} \[ -\frac {\sin ^2\left (a+\frac {b}{x}\right )}{2 b^2 x}-\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{4 b^3}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^2}+\frac {1}{4 b^2 x}-\frac {1}{6 x^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2635
Rule 3311
Rule 3379
Rubi steps
\begin {align*} \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 \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^2}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{2 b^2 x}-\frac {1}{2} \operatorname {Subst}\left (\int x^2 \, dx,x,\frac {1}{x}\right )+\frac {\operatorname {Subst}\left (\int \sin ^2(a+b x) \, dx,x,\frac {1}{x}\right )}{2 b^2}\\ &=-\frac {1}{6 x^3}-\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{4 b^3}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^2}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{2 b^2 x}+\frac {\operatorname {Subst}\left (\int 1 \, dx,x,\frac {1}{x}\right )}{4 b^2}\\ &=-\frac {1}{6 x^3}+\frac {1}{4 b^2 x}-\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{4 b^3}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^2}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{2 b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 54, normalized size = 0.62 \[ \frac {-3 \left (x^3-2 b^2 x\right ) \sin \left (2 \left (a+\frac {b}{x}\right )\right )+6 b x^2 \cos \left (2 \left (a+\frac {b}{x}\right )\right )-4 b^3}{24 b^3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 72, normalized size = 0.83 \[ \frac {6 \, b x^{2} \cos \left (\frac {a x + b}{x}\right )^{2} - 2 \, b^{3} - 3 \, b x^{2} + 3 \, {\left (2 \, b^{2} x - x^{3}\right )} \cos \left (\frac {a x + b}{x}\right ) \sin \left (\frac {a x + b}{x}\right )}{12 \, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 153, normalized size = 1.76 \[ \frac {6 \, a^{2} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {12 \, {\left (a x + b\right )} a^{2}}{x} - 6 \, a \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {12 \, {\left (a x + b\right )} a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {12 \, {\left (a x + b\right )}^{2} a}{x^{2}} + \frac {6 \, {\left (a x + b\right )} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {6 \, {\left (a x + b\right )}^{2} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} - \frac {4 \, {\left (a x + b\right )}^{3}}{x^{3}} - 3 \, \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 197, normalized size = 2.26 \[ -\frac {\left (a +\frac {b}{x}\right )^{2} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right ) \left (\cos ^{2}\left (a +\frac {b}{x}\right )\right )}{2}+\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{4}+\frac {b}{4 x}+\frac {a}{4}-\frac {\left (a +\frac {b}{x}\right )^{3}}{3}-2 a \left (\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{4}\right )+a^{2} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 68, normalized size = 0.78 \[ -\frac {{\left ({\left (3 i \, \Gamma \left (3, \frac {2 i \, b}{x}\right ) - 3 i \, \Gamma \left (3, -\frac {2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) + 3 \, {\left (\Gamma \left (3, \frac {2 i \, b}{x}\right ) + \Gamma \left (3, -\frac {2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right )\right )} x^{3} + 16 \, b^{3}}{96 \, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.70, size = 64, normalized size = 0.74 \[ \frac {\frac {b\,x^2\,\cos \left (2\,a+\frac {2\,b}{x}\right )}{4}-\frac {b^3}{6}+\frac {b^2\,x\,\sin \left (2\,a+\frac {2\,b}{x}\right )}{4}}{b^3\,x^3}-\frac {\sin \left (2\,a+\frac {2\,b}{x}\right )}{8\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.97, size = 654, normalized size = 7.52 \[ \begin {cases} - \frac {2 b^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} - \frac {4 b^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} - \frac {2 b^{3}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} - \frac {12 b^{2} x \tan ^{3}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} + \frac {12 b^{2} x \tan {\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} + \frac {3 b x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} - \frac {18 b x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} + \frac {3 b x^{2}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} + \frac {6 x^{3} \tan ^{3}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} - \frac {6 x^{3} \tan {\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{12 b^{3} x^{3} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 24 b^{3} x^{3} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 12 b^{3} x^{3}} & \text {for}\: b \neq 0 \\- \frac {\sin ^{2}{\relax (a )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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