Optimal. Leaf size=65 \[ b^2 (-\cos (2 a)) \text {Ci}\left (\frac {2 b}{x}\right )+b^2 \sin (2 a) \text {Si}\left (\frac {2 b}{x}\right )+\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b x \sin \left (2 \left (a+\frac {b}{x}\right )\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3393, 4573, 3373, 3361, 3297, 3303, 3299, 3302} \[ b^2 (-\cos (2 a)) \text {CosIntegral}\left (\frac {2 b}{x}\right )+b^2 \sin (2 a) \text {Si}\left (\frac {2 b}{x}\right )+\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b x \sin \left (2 \left (a+\frac {b}{x}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3361
Rule 3373
Rule 3393
Rule 4573
Rubi steps
\begin {align*} \int x \sin ^2\left (a+\frac {b}{x}\right ) \, dx &=\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+b \int \cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right ) \, dx\\ &=\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b \int \sin \left (2 \left (a+\frac {b}{x}\right )\right ) \, dx\\ &=\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b \int \sin \left (2 a+\frac {2 b}{x}\right ) \, dx\\ &=\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\sin (2 a+2 b x)}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b x \sin \left (2 \left (a+\frac {b}{x}\right )\right )-b^2 \operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b x \sin \left (2 \left (a+\frac {b}{x}\right )\right )-\left (b^2 \cos (2 a)\right ) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{x} \, dx,x,\frac {1}{x}\right )+\left (b^2 \sin (2 a)\right ) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=-b^2 \cos (2 a) \text {Ci}\left (\frac {2 b}{x}\right )+\frac {1}{2} x^2 \sin ^2\left (a+\frac {b}{x}\right )+\frac {1}{2} b x \sin \left (2 \left (a+\frac {b}{x}\right )\right )+b^2 \sin (2 a) \text {Si}\left (\frac {2 b}{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 65, normalized size = 1.00 \[ b^2 (-\cos (2 a)) \text {Ci}\left (\frac {2 b}{x}\right )+b^2 \sin (2 a) \text {Si}\left (\frac {2 b}{x}\right )+\frac {1}{4} x \left (2 b \sin \left (2 \left (a+\frac {b}{x}\right )\right )+x \left (-\cos \left (2 \left (a+\frac {b}{x}\right )\right )\right )+x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 90, normalized size = 1.38 \[ -\frac {1}{2} \, x^{2} \cos \left (\frac {a x + b}{x}\right )^{2} + b x \cos \left (\frac {a x + b}{x}\right ) \sin \left (\frac {a x + b}{x}\right ) + b^{2} \sin \left (2 \, a\right ) \operatorname {Si}\left (\frac {2 \, b}{x}\right ) + \frac {1}{2} \, x^{2} - \frac {1}{2} \, {\left (b^{2} \operatorname {Ci}\left (\frac {2 \, b}{x}\right ) + b^{2} \operatorname {Ci}\left (-\frac {2 \, b}{x}\right )\right )} \cos \left (2 \, a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 283, normalized size = 4.35 \[ -\frac {4 \, a^{2} b^{3} \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, a + \frac {2 \, {\left (a x + b\right )}}{x}\right ) + 4 \, a^{2} b^{3} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, a - \frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {8 \, {\left (a x + b\right )} a b^{3} \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, a + \frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} - \frac {8 \, {\left (a x + b\right )} a b^{3} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, a - \frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {4 \, {\left (a x + b\right )}^{2} b^{3} \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, a + \frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} + 2 \, a b^{3} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) + \frac {4 \, {\left (a x + b\right )}^{2} b^{3} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, a - \frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} + b^{3} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {2 \, {\left (a x + b\right )} b^{3} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} - b^{3}}{4 \, {\left (a^{2} - \frac {2 \, {\left (a x + b\right )} a}{x} + \frac {{\left (a x + b\right )}^{2}}{x^{2}}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.17 \[ -b^{2} \left (-\frac {x^{2}}{4 b^{2}}+\frac {\cos \left (2 a +\frac {2 b}{x}\right ) x^{2}}{4 b^{2}}-\frac {\sin \left (2 a +\frac {2 b}{x}\right ) x}{2 b}-\Si \left (\frac {2 b}{x}\right ) \sin \left (2 a \right )+\Ci \left (\frac {2 b}{x}\right ) \cos \left (2 a \right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 89, normalized size = 1.37 \[ -\frac {1}{4} \, {\left (2 \, {\left ({\rm Ei}\left (\frac {2 i \, b}{x}\right ) + {\rm Ei}\left (-\frac {2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) - {\left (-2 i \, {\rm Ei}\left (\frac {2 i \, b}{x}\right ) + 2 i \, {\rm Ei}\left (-\frac {2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right )\right )} b^{2} - \frac {1}{4} \, x^{2} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) + \frac {1}{2} \, b x \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) + \frac {1}{4} \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\sin \left (a+\frac {b}{x}\right )}^2 \,d x \]
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 \[ \int x \sin ^{2}{\left (a + \frac {b}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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