Optimal. Leaf size=185 \[ -\frac {15 x^2}{4 b^5 \pi ^3}+\frac {x^6}{12 b \pi }+\frac {7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^2 \pi }-\frac {15 \text {FresnelC}(b x)^2}{2 b^7 \pi ^3}+\frac {5 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {11 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6598, 6590,
6576, 30, 3461, 2714, 3460, 3377, 2717, 3390} \begin {gather*} -\frac {15 \text {FresnelC}(b x)^2}{2 \pi ^3 b^7}-\frac {15 x^2}{4 \pi ^3 b^5}-\frac {x^5 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {11 \sin \left (\pi b^2 x^2\right )}{2 \pi ^4 b^7}+\frac {15 x \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {7 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac {5 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^4 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {x^6}{12 \pi b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2714
Rule 2717
Rule 3377
Rule 3390
Rule 3460
Rule 3461
Rule 6576
Rule 6590
Rule 6598
Rubi steps
\begin {align*} \int x^6 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {5 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^2 \pi }+\frac {\int x^5 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {5 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {15 \int x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac {5 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}+\frac {\text {Subst}\left (\int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi }\\ &=\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {5 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {15 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^6 \pi ^3}-\frac {15 \int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {5 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}+\frac {\text {Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b \pi }+\frac {\text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=\frac {x^6}{12 b \pi }+\frac {5 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {5 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {15 \text {Subst}(\int x \, dx,x,C(b x))}{b^7 \pi ^3}-\frac {5 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {15 \text {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}-\frac {\text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=-\frac {15 x^2}{4 b^5 \pi ^3}+\frac {x^6}{12 b \pi }+\frac {7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {15 C(b x)^2}{2 b^7 \pi ^3}+\frac {5 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {5 \sin \left (b^2 \pi x^2\right )}{b^7 \pi ^4}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}\\ &=-\frac {15 x^2}{4 b^5 \pi ^3}+\frac {x^6}{12 b \pi }+\frac {7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {15 C(b x)^2}{2 b^7 \pi ^3}+\frac {5 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {11 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 185, normalized size = 1.00 \begin {gather*} -\frac {15 x^2}{4 b^5 \pi ^3}+\frac {x^6}{12 b \pi }+\frac {7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {15 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^6 \pi ^3}-\frac {x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^2 \pi }-\frac {15 \text {FresnelC}(b x)^2}{2 b^7 \pi ^3}+\frac {5 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {11 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{6} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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 = 0.37, size = 141, normalized size = 0.76 \begin {gather*} \frac {\pi ^{3} b^{6} x^{6} + 42 \, \pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 66 \, \pi b^{2} x^{2} - 12 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 90 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (10 \, \pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) + {\left (\pi ^{2} b^{4} x^{4} - 22\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{12 \, \pi ^{4} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.64, size = 264, normalized size = 1.43 \begin {gather*} \begin {cases} \frac {x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{12 \pi b} + \frac {x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{12 \pi b} - \frac {x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} + \frac {x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} + \frac {5 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {11 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{3} b^{5}} - \frac {2 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} + \frac {15 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} - \frac {11 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} - \frac {15 C^{2}\left (b x\right )}{2 \pi ^{3} b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^6\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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