Optimal. Leaf size=56 \[ \frac{\sin (a x)}{a^2 x^2}+\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text{Si}(a x)+\frac{\cos (a x)}{a x} \]
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Rubi [A] time = 0.101601, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {4598, 3297, 3299} \[ \frac{\sin (a x)}{a^2 x^2}+\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text{Si}(a x)+\frac{\cos (a x)}{a x} \]
Antiderivative was successfully verified.
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Rule 4598
Rule 3297
Rule 3299
Rubi steps
\begin{align*} \int \frac{\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx &=\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}-\frac{2 \int \frac{\sin (a x)}{x^3} \, dx}{a^2}\\ &=\frac{\sin (a x)}{a^2 x^2}+\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}-\frac{\int \frac{\cos (a x)}{x^2} \, dx}{a}\\ &=\frac{\cos (a x)}{a x}+\frac{\sin (a x)}{a^2 x^2}+\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\int \frac{\sin (a x)}{x} \, dx\\ &=\frac{\cos (a x)}{a x}+\frac{\sin (a x)}{a^2 x^2}+\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text{Si}(a x)\\ \end{align*}
Mathematica [C] time = 7.54569, size = 242, normalized size = 4.32 \[ \frac{-i e \text{CosIntegral}(-a x+i) (a x \cos (a x)-\sin (a x))+i e \text{CosIntegral}(a x+i) (a x \cos (a x)-\sin (a x))-i e \text{ExpIntegralEi}(-1-i a x) \sin (a x)+i e \text{ExpIntegralEi}(-1+i a x) \sin (a x)+i e a x \text{ExpIntegralEi}(-1-i a x) \cos (a x)-i e a x \text{ExpIntegralEi}(-1+i a x) \cos (a x)-2 \text{Si}(a x) \sin (a x)-e \text{Si}(i-a x) \sin (a x)+e \text{Si}(a x+i) \sin (a x)+2 a x \text{Si}(a x) \cos (a x)+e a x \text{Si}(i-a x) \cos (a x)-e a x \text{Si}(a x+i) \cos (a x)+\cos (2 a x)+1}{2 a x \cos (a x)-2 \sin (a x)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.666, size = 108, normalized size = 1.9 \begin{align*}{\frac{{\frac{i}{2}}{{\rm e}^{iax}}}{-1+iax}}+{\frac{i}{2}}{\it Ei} \left ( 1,-iax \right ) +{\frac{{\frac{i}{2}}{{\rm e}^{-iax}}}{1+iax}}-{\frac{i}{2}}{\it Ei} \left ( 1,iax \right ) +2\,{\frac{{{\rm e}^{iax}}}{ \left ( ax+i \right ) \left ( ax-i \right ) \left ( ax{{\rm e}^{2\,iax}}+i{{\rm e}^{2\,iax}}+ax-i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08375, size = 138, normalized size = 2.46 \begin{align*} \frac{a x \cos \left (a x\right ) \operatorname{Si}\left (a x\right ) + \cos \left (a x\right )^{2} - \sin \left (a x\right ) \operatorname{Si}\left (a x\right )}{a x \cos \left (a x\right ) - \sin \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.28642, size = 670, normalized size = 11.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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