Optimal. Leaf size=124 \[ -\frac {105 C(b x)}{8 \pi ^4 b^8}-\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi b}+\frac {105 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac {35 x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac {7 x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac {1}{8} x^8 C(b x) \]
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Rubi [A] time = 0.08, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6427, 3386, 3385, 3352} \[ -\frac {105 \text {FresnelC}(b x)}{8 \pi ^4 b^8}-\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi b}+\frac {35 x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac {7 x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac {105 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac {1}{8} x^8 \text {FresnelC}(b x) \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3385
Rule 3386
Rule 6427
Rubi steps
\begin {align*} \int x^7 C(b x) \, dx &=\frac {1}{8} x^8 C(b x)-\frac {1}{8} b \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{8} x^8 C(b x)-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {7 \int x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b \pi }\\ &=-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {35 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^3 \pi ^2}\\ &=-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac {105 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}\\ &=\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac {105 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^7 \pi ^4}\\ &=\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {105 C(b x)}{8 b^8 \pi ^4}+\frac {1}{8} x^8 C(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }\\ \end {align*}
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Mathematica [A] time = 0.08, size = 89, normalized size = 0.72 \[ \frac {\left (\pi ^4 b^8 x^8-105\right ) C(b x)-7 b x \left (\pi ^2 b^4 x^4-15\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )+\pi b^3 x^3 \left (35-\pi ^2 b^4 x^4\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^8} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{7} {\rm fresnelc}\left (b x\right ), x\right ) \]
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 \[ \int x^{7} {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 123, normalized size = 0.99 \[ \frac {\frac {\FresnelC \left (b x \right ) b^{8} x^{8}}{8}-\frac {b^{7} x^{7} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {-\frac {7 b^{5} x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {7 \left (\frac {5 b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {15 \left (-\frac {b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\FresnelC \left (b x \right )}{\pi }\right )}{\pi }\right )}{8 \pi }}{\pi }}{b^{8}} \]
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 \[ \int x^{7} {\rm fresnelc}\left (b x\right )\,{d x} \]
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 \[ \int x^7\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.31, size = 184, normalized size = 1.48 \[ \frac {45 x^{8} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{512 \Gamma \left (\frac {13}{4}\right )} - \frac {45 x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi b \Gamma \left (\frac {13}{4}\right )} - \frac {315 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{2} b^{3} \Gamma \left (\frac {13}{4}\right )} + \frac {1575 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{3} b^{5} \Gamma \left (\frac {13}{4}\right )} + \frac {4725 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{4} b^{7} \Gamma \left (\frac {13}{4}\right )} - \frac {4725 C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{4} b^{8} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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