Optimal. Leaf size=231 \[ \frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \text {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6590, 6598,
6576, 30, 3461, 2714, 3460, 3377, 2717, 3390} \begin {gather*} \frac {105 \text {FresnelC}(b x)^2}{2 \pi ^4 b^9}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {7 x^6}{12 \pi ^2 b^3}+\frac {x^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {35 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 \int x^6 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {35 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^4 \pi ^2}-\frac {7 \int x^5 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\text {Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {105 \int x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}+\frac {35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}-\frac {3 \text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {7 \text {Subst}\left (\int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 x^4 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {105 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^8 \pi ^4}+\frac {105 \int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}+\frac {3 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac {35 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {7 \text {Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {7 \text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}\\ &=-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {41 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {105 \text {Subst}(\int x \, dx,x,C(b x))}{b^9 \pi ^4}+\frac {3 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {35 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}+\frac {105 \text {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {7 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}\\ &=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {105 C(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {73 \sin \left (b^2 \pi x^2\right )}{2 b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {7 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}\\ &=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {105 C(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 231, normalized size = 1.00 \begin {gather*} \frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \text {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int x^{8} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 169, normalized size = 0.73 \begin {gather*} -\frac {5 \, \pi ^{3} b^{6} x^{6} - 240 \, \pi b^{2} x^{2} - 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 42 \, {\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 ) - 315 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{6 \, \pi ^{5} b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 16.70, size = 301, normalized size = 1.30 \begin {gather*} \begin {cases} \frac {x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} - \frac {5 x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{6 \pi ^{2} b^{3}} - \frac {x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{3 \pi ^{2} b^{3}} + \frac {7 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {5 x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac {35 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} + \frac {40 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} + \frac {25 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{4} b^{7}} - \frac {105 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{4} b^{8}} + \frac {80 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{5} b^{9}} + \frac {105 C^{2}\left (b x\right )}{2 \pi ^{4} b^{9}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^8\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________