Optimal. Leaf size=140 \[ \frac {3 x^2}{8 b^2 \pi ^2}-\frac {x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{2 b^3 \pi ^2}+\frac {3 \text {FresnelC}(b x)^2}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \text {FresnelC}(b x)^2-\frac {x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{2 b^4 \pi ^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6566, 6590,
6598, 6576, 30, 3461, 2714, 3460, 3377, 2717} \begin {gather*} \frac {3 \text {FresnelC}(b x)^2}{4 \pi ^2 b^4}-\frac {x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac {3 x^2}{8 \pi ^2 b^2}-\frac {x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac {\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}-\frac {3 x \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac {1}{4} x^4 \text {FresnelC}(b x)^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2714
Rule 2717
Rule 3377
Rule 3460
Rule 3461
Rule 6566
Rule 6576
Rule 6590
Rule 6598
Rubi steps
\begin {align*} \int x^3 C(b x)^2 \, dx &=\frac {1}{4} x^4 C(b x)^2-\frac {1}{2} b \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {1}{4} x^4 C(b x)^2-\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {\int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{4 \pi }+\frac {3 \int x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{2 b^3 \pi ^2}+\frac {1}{4} x^4 C(b x)^2-\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {3 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{2 b^3 \pi ^2}+\frac {3 \int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{2 b^2 \pi ^2}+\frac {\text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 \pi }\\ &=-\frac {x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{2 b^3 \pi ^2}+\frac {1}{4} x^4 C(b x)^2-\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {3 \text {Subst}(\int x \, dx,x,C(b x))}{2 b^4 \pi ^2}+\frac {\text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^2 \pi ^2}+\frac {3 \text {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^2 \pi ^2}\\ &=\frac {3 x^2}{8 b^2 \pi ^2}-\frac {x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{2 b^3 \pi ^2}+\frac {3 C(b x)^2}{4 b^4 \pi ^2}+\frac {1}{4} x^4 C(b x)^2-\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{2 b^4 \pi ^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 140, normalized size = 1.00 \begin {gather*} \frac {3 x^2}{8 b^2 \pi ^2}-\frac {x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{2 b^3 \pi ^2}+\frac {3 \text {FresnelC}(b x)^2}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \text {FresnelC}(b x)^2-\frac {x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{2 b^4 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int x^{3} \FresnelC \left (b x \right )^{2}\, dx\]
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 = 118, normalized size = 0.84 \begin {gather*} -\frac {\pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, \pi b^{2} x^{2} + 6 \, \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - {\left (3 \, \pi + \pi ^{3} b^{4} x^{4}\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, \pi ^{3} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} C^{2}\left (b x\right )\, dx \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 = -0.01 \begin {gather*} \int x^3\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________