3.2.66 \(\int \frac {\text {FresnelC}(d (a+b \log (c x^n)))}{x} \, dx\) [166]

Optimal. Leaf size=66 \[ \frac {\text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{b d n \pi } \]

[Out]

FresnelC(d*(a+b*ln(c*x^n)))*(a+b*ln(c*x^n))/b/n-sin(1/2*d^2*Pi*(a+b*ln(c*x^n))^2)/b/d/n/Pi

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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6554} \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right ) \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\sin \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )\right )^2\right )}{\pi b d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[FresnelC[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(FresnelC[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n) - Sin[(d^2*Pi*(a + b*Log[c*x^n])^2)/2]/(b*d*n*Pi)

Rule 6554

Int[FresnelC[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(FresnelC[a + b*x]/b), x] - Simp[Sin[(Pi/2)*(a +
b*x)^2]/(b*Pi), x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {align*} \int \frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int C(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {Subst}\left (\int C(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=\frac {C\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin \left (\frac {1}{2} \pi \left (a d+b d \log \left (c x^n\right )\right )^2\right )}{b d n \pi }\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(66)=132\).
time = 0.07, size = 165, normalized size = 2.50 \begin {gather*} \frac {a \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n}-\frac {\cos \left (a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \sin \left (\frac {1}{2} a^2 d^2 \pi \right )}{b d n \pi }-\frac {\cos \left (\frac {1}{2} a^2 d^2 \pi \right ) \sin \left (a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{b d n \pi } \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(a*FresnelC[d*(a + b*Log[c*x^n])])/(b*n) + (FresnelC[d*(a + b*Log[c*x^n])]*Log[c*x^n])/n - (Cos[a*b*d^2*Pi*Log
[c*x^n] + (b^2*d^2*Pi*Log[c*x^n]^2)/2]*Sin[(a^2*d^2*Pi)/2])/(b*d*n*Pi) - (Cos[(a^2*d^2*Pi)/2]*Sin[a*b*d^2*Pi*L
og[c*x^n] + (b^2*d^2*Pi*Log[c*x^n]^2)/2])/(b*d*n*Pi)

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Maple [A]
time = 1.47, size = 64, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {\FresnelC \left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\frac {\sin \left (\frac {\pi \left (a d +b d \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{\pi }}{n b d}\)	\(64\)
default	\(\frac {\FresnelC \left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\frac {\sin \left (\frac {\pi \left (a d +b d \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{\pi }}{n b d}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)

[Out]

1/n/b/d*(FresnelC(a*d+b*d*ln(c*x^n))*(a*d+b*d*ln(c*x^n))-1/Pi*sin(1/2*Pi*(a*d+b*d*ln(c*x^n))^2))

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Maxima [A]
time = 0.26, size = 82, normalized size = 1.24 \begin {gather*} \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} d^{2} \log \left (c x^{n}\right )^{2} + \pi a b d^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi }}{b d n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

((b*log(c*x^n) + a)*d*fresnel_cos((b*log(c*x^n) + a)*d) - sin(1/2*pi*b^2*d^2*log(c*x^n)^2 + pi*a*b*d^2*log(c*x
^n) + 1/2*pi*a^2*d^2)/pi)/(b*d*n)

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Fricas [A]
time = 0.37, size = 121, normalized size = 1.83 \begin {gather*} \frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sin \left (\frac {1}{2} \, \pi b^{2} d^{2} n^{2} \log \left (x\right )^{2} + \pi b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, \pi b^{2} d^{2} \log \left (c\right )^{2} + \pi a b d^{2} n \log \left (x\right ) + \pi a b d^{2} \log \left (c\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi b d n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")

[Out]

((pi*b*d*n*log(x) + pi*b*d*log(c) + pi*a*d)*fresnel_cos(b*d*log(c*x^n) + a*d) - sin(1/2*pi*b^2*d^2*n^2*log(x)^
2 + pi*b^2*d^2*n*log(c)*log(x) + 1/2*pi*b^2*d^2*log(c)^2 + pi*a*b*d^2*n*log(x) + pi*a*b*d^2*log(c) + 1/2*pi*a^
2*d^2))/(pi*b*d*n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {C\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(fresnelc(a*d + b*d*log(c*x**n))/x, x)

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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]

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")

[Out]

integrate(fresnel_cos((b*log(c*x^n) + a)*d)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(d*(a + b*log(c*x^n)))/x,x)

[Out]

int(FresnelC(d*(a + b*log(c*x^n)))/x, x)

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