Optimal. Leaf size=37 \[ \frac {(a+b x) \text {FresnelC}(a+b x)}{b}-\frac {\sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi } \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6554}
\begin {gather*} \frac {(a+b x) \text {FresnelC}(a+b x)}{b}-\frac {\sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6554
Rubi steps
\begin {align*} \int C(a+b x) \, dx &=\frac {(a+b x) C(a+b x)}{b}-\frac {\sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi }\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(37)=74\).
time = 0.02, size = 90, normalized size = 2.43 \begin {gather*} \frac {a \text {FresnelC}(a+b x)}{b}+x \text {FresnelC}(a+b x)-\frac {\cos \left (a b \pi x+\frac {1}{2} b^2 \pi x^2\right ) \sin \left (\frac {a^2 \pi }{2}\right )}{b \pi }-\frac {\cos \left (\frac {a^2 \pi }{2}\right ) \sin \left (a b \pi x+\frac {1}{2} b^2 \pi x^2\right )}{b \pi } \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 34, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\FresnelC \left (b x +a \right ) \left (b x +a \right )-\frac {\sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\) | \(34\) |
default | \(\frac {\FresnelC \left (b x +a \right ) \left (b x +a \right )-\frac {\sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 44, normalized size = 1.19 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {C}\left (b x + a\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi }}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 47, normalized size = 1.27 \begin {gather*} \frac {{\left (\pi b x + \pi a\right )} \operatorname {C}\left (b x + a\right ) - \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi b} \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 C\left (a + 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 = -0.03 \begin {gather*} \int \mathrm {FresnelC}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________