Optimal. Leaf size=231 \[ \frac {105 C(b x)^2}{2 \pi ^4 b^9}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {7 x^6}{12 \pi ^2 b^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.38, 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 = {6455, 6463, 6441, 30, 3380, 2634, 3379, 3296, 2637, 3309} \[ \frac {x^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {35 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {7 x^5 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {105 x \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac {105 \text {FresnelC}(b x)^2}{2 \pi ^4 b^9}-\frac {7 x^6}{12 \pi ^2 b^3}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2634
Rule 2637
Rule 3296
Rule 3309
Rule 3379
Rule 3380
Rule 6441
Rule 6455
Rule 6463
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 {\operatorname {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 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {7 \operatorname {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 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac {35 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {7 \operatorname {Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {7 \operatorname {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 \operatorname {Subst}(\int x \, dx,x,C(b x))}{b^9 \pi ^4}+\frac {3 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {35 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}+\frac {105 \operatorname {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 \operatorname {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 \operatorname {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 \[ \frac {105 C(b x)^2}{2 \pi ^4 b^9}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {7 x^6}{12 \pi ^2 b^3}+\frac {x^7 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, 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 \[ \int x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^8\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________