Optimal. Leaf size=9 \[ \frac {\log (\text {FresnelC}(b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6576, 29}
\begin {gather*} \frac {\log (\text {FresnelC}(b x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 6576
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{C(b x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,C(b x)\right )}{b}\\ &=\frac {\log (C(b x))}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log (\text {FresnelC}(b x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.23, size = 10, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\ln \left (\FresnelC \left (b x \right )\right )}{b}\) | \(10\) |
default | \(\frac {\ln \left (\FresnelC \left (b x \right )\right )}{b}\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\operatorname {C}\left (b x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\operatorname {C}\left (b x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 10, normalized size = 1.11 \begin {gather*} \begin {cases} \frac {\log {\left (C\left (b x\right ) \right )}}{b} & \text {for}\: b \neq 0 \\\tilde {\infty } x & \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 = -0.11 \begin {gather*} \int \frac {\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{\mathrm {FresnelC}\left (b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________