Optimal. Leaf size=89 \[ \frac {1}{12} b^3 \pi \text {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}+\frac {1}{3} b^2 \pi \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^2},x\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx &=-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{6} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{12} b \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx+\frac {1}{12} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=\frac {1}{12} b^3 \pi \text {Ci}\left (b^2 \pi x^2\right )-\frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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