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 = 0.10, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 3297
Rule 3299
Rule 4598
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*}
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Mathematica [C] time = 7.42, size = 242, normalized size = 4.32 \[ \frac {-i e \text {Ci}(i-a x) (a x \cos (a x)-\sin (a x))+i e \text {Ci}(a x+i) (a x \cos (a x)-\sin (a x))-i e \text {Ei}(-i a x-1) \sin (a x)+i e \text {Ei}(i a x-1) \sin (a x)+i e a x \text {Ei}(-i a x-1) \cos (a x)-i e a x \text {Ei}(i a x-1) \cos (a x)-2 \text {Si}(a x) \sin (a x)-e \text {Si}(i-a x) \sin (a x)+e \text {Si}(a x+i) \sin (a x)+2 a x \text {Si}(a x) \cos (a x)+e a x \text {Si}(i-a x) \cos (a x)-e a x \text {Si}(a x+i) \cos (a x)+\cos (2 a x)+1}{2 a x \cos (a x)-2 \sin (a x)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.87, size = 45, normalized size = 0.80 \[ \frac {a x \cos \left (a x\right ) \operatorname {Si}\left (a x\right ) + \cos \left (a x\right )^{2} - \sin \left (a x\right ) \operatorname {Si}\left (a x\right )}{a x \cos \left (a x\right ) - \sin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.37, size = 496, normalized size = 8.86 \[ \frac {a^{3} x^{3} \Im \left (\operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} - a^{3} x^{3} \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{3} x^{3} \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \Im \left (\operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 2 \, a^{2} x^{2} \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + 4 \, a^{2} x^{2} \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 2 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{4} - a^{3} x^{3} \Im \left (\operatorname {Ci}\left (a x\right ) \right ) + a^{3} x^{3} \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) - 2 \, a^{3} x^{3} \operatorname {Si}\left (a x\right ) + a x \Im \left (\operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} - a x \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a x \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \Im \left (\operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) - 2 \, a^{2} x^{2} \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) + 4 \, a^{2} x^{2} \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right ) + 4 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{2} - 2 \, a^{2} x^{2} + 2 \, \Im \left (\operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 2 \, \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + 4 \, \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 4 \, \tan \left (\frac {1}{2} \, a x\right )^{4} - a x \Im \left (\operatorname {Ci}\left (a x\right ) \right ) + a x \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) - 2 \, a x \operatorname {Si}\left (a x\right ) + 2 \, \Im \left (\operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) - 2 \, \Im \left (\operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) + 4 \, \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right ) - 4}{2 \, {\left (a^{3} x^{3} \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{3} - a^{3} x^{3} + a x \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right ) + 2 \, \tan \left (\frac {1}{2} \, a x\right )^{3} - a x + 2 \, \tan \left (\frac {1}{2} \, a x\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.28, size = 108, normalized size = 1.93 \[ \frac {i \Ei \left (1, -i a x \right )}{2}+\frac {i {\mathrm e}^{i a x}}{2 i a x -2}+\frac {i {\mathrm e}^{-i a x}}{2 i a x +2}-\frac {i \Ei \left (1, i a x \right )}{2}+\frac {2 \,{\mathrm e}^{i a x}}{\left (a x +i\right ) \left (a x -i\right ) \left (a x \,{\mathrm e}^{2 i a x}+i {\mathrm e}^{2 i a x}+a x -i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\sin \left (a\,x\right )}^3}{x\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\sin ^{3}{\left (a x \right )}}{x \left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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