3.2.69 \(\int (e x)^m \text {FresnelC}(d (a+b \log (c x^n))) \, dx\) [169]

Optimal. Leaf size=280 \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m-i a b d^2 n \pi -i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {(e x)^{1+m} \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)} \]

[Out]

(1/4+1/4*I)*exp(1/2*I*(1+m)*(1+m+2*I*a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)*x*(e*x)^m*erf((1/2+1/2*I)*(1+m+I*a*b*d^2*n*
Pi+I*b^2*d^2*n*Pi*ln(c*x^n))/b/d/n/Pi^(1/2))/(1+m)/((c*x^n)^((1+m)/n))-(1/4+1/4*I)*x*(e*x)^m*erfi((1/2+1/2*I)*
(1+m-I*a*b*d^2*n*Pi-I*b^2*d^2*n*Pi*ln(c*x^n))/b/d/n/Pi^(1/2))/exp(1/2*I*(1+m)*(1+m-2*I*a*b*d^2*n*Pi)/b^2/d^2/n
^2/Pi)/(1+m)/((c*x^n)^((1+m)/n))+(e*x)^(1+m)*FresnelC(d*(a+b*ln(c*x^n)))/e/(1+m)

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Rubi [A]
time = 0.37, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6607, 4714, 2314, 2308, 2266, 2235, 2236} \begin {gather*} \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (\frac {i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{m+1}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (-\frac {i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2 n-i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{m+1}+\frac {(e x)^{m+1} \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1/4 + I/4)*E^(((I/2)*(1 + m)*(1 + m + (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*x*(e*x)^m*Erf[((1/2 + I/2)*(1 +
m + I*a*b*d^2*n*Pi + I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/((1 + m)*(c*x^n)^((1 + m)/n)) - ((1/4 + I
/4)*x*(e*x)^m*Erfi[((1/2 + I/2)*(1 + m - I*a*b*d^2*n*Pi - I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/(E^((
(I/2)*(1 + m)*(1 + m - (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*(1 + m)*(c*x^n)^((1 + m)/n)) + ((e*x)^(1 + m)*Fr
esnelC[d*(a + b*Log[c*x^n])])/(e*(1 + m))

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2308

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n)), Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Lo
g[F]*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]

Rule 2314

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^m*((c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(m + 2*a*b*f*n*Log[F])), Int[(d + e*x)^(m + 2
*a*b*f*n*Log[F])*F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x]
&& EqQ[e*g - d*h, 0]

Rule 4714

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m/
E^(I*d*(a + b*Log[c*x^n])^2), x], x] + Dist[1/2, Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x]

Rule 6607

Int[FresnelC[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*
(FresnelC[d*(a + b*Log[c*x^n])]/(e*(m + 1))), x] - Dist[b*d*(n/(m + 1)), Int[(e*x)^m*Cos[(Pi/2)*(d*(a + b*Log[
c*x^n]))^2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (e x)^m C\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int (e x)^m \cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx}{1+m}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{2 (1+m)}-\frac {(b d n) \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{2 (1+m)}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -i a b d^2 \pi \log \left (c x^n\right )-\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) (e x)^m \, dx}{2 (1+m)}-\frac {(b d n) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +i a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) (e x)^m \, dx}{2 (1+m)}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) (e x)^m \left (c x^n\right )^{-i a b d^2 \pi } \, dx}{2 (1+m)}-\frac {(b d n) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) (e x)^m \left (c x^n\right )^{i a b d^2 \pi } \, dx}{2 (1+m)}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (b d n x^{i a b d^2 n \pi } \left (c x^n\right )^{-i a b d^2 \pi }\right ) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{-i a b d^2 n \pi } (e x)^m \, dx}{2 (1+m)}-\frac {\left (b d n x^{-i a b d^2 n \pi } \left (c x^n\right )^{i a b d^2 \pi }\right ) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{i a b d^2 n \pi } (e x)^m \, dx}{2 (1+m)}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (b d n x^{-m+i a b d^2 n \pi } (e x)^m \left (c x^n\right )^{-i a b d^2 \pi }\right ) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{m-i a b d^2 n \pi } \, dx}{2 (1+m)}-\frac {\left (b d n x^{-m-i a b d^2 n \pi } (e x)^m \left (c x^n\right )^{i a b d^2 \pi }\right ) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{m+i a b d^2 n \pi } \, dx}{2 (1+m)}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (b d x (e x)^m \left (c x^n\right )^{-i a b d^2 \pi -\frac {1+m-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {1}{2} i a^2 d^2 \pi +\frac {\left (1+m-i a b d^2 n \pi \right ) x}{n}-\frac {1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}-\frac {\left (b d x (e x)^m \left (c x^n\right )^{i a b d^2 \pi -\frac {1+m+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {\left (1+m+i a b d^2 n \pi \right ) x}{n}+\frac {1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}\\ &=\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (b d \exp \left (-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{-i a b d^2 \pi -\frac {1+m-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {i \left (\frac {1+m-i a b d^2 n \pi }{n}-i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}-\frac {\left (b d \exp \left (\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{i a b d^2 \pi -\frac {1+m+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {i \left (\frac {1+m+i a b d^2 n \pi }{n}+i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}\\ &=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \exp \left (\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \exp \left (-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m-i a b d^2 n \pi -i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {(e x)^{1+m} C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}\\ \end {align*}

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Mathematica [A]
time = 3.50, size = 244, normalized size = 0.87 \begin {gather*} \frac {(e x)^m \left ((-1)^{3/4} \sqrt {2} e^{-\frac {(1+m) \left (i+i m+2 a b d^2 n \pi +2 b^2 d^2 n \pi \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{2 b^2 d^2 n^2 \pi }} x^{-m} \left (\text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i+i m+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )-e^{\frac {i (1+m)^2}{b^2 d^2 n^2 \pi }} \text {Erfi}\left (\frac {(-1)^{3/4} \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right )+4 x \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4 (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^m*(((-1)^(3/4)*Sqrt[2]*(Erf[((1/2 + I/2)*(I + I*m + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqr
t[Pi])] - E^((I*(1 + m)^2)/(b^2*d^2*n^2*Pi))*Erfi[((-1)^(3/4)*(1 + m + I*a*b*d^2*n*Pi + I*b^2*d^2*n*Pi*Log[c*x
^n]))/(b*d*n*Sqrt[2*Pi])]))/(E^(((1 + m)*(I + I*m + 2*a*b*d^2*n*Pi + 2*b^2*d^2*n*Pi*(-(n*Log[x]) + Log[c*x^n])
))/(2*b^2*d^2*n^2*Pi))*x^m) + 4*x*FresnelC[d*(a + b*Log[c*x^n])]))/(4*(1 + m))

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*FresnelC(d*(a+b*ln(c*x^n))),x)

[Out]

int((e*x)^m*FresnelC(d*(a+b*ln(c*x^n))),x)

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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]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (312) = 624\).
time = 0.39, size = 674, normalized size = 2.41 \begin {gather*} -\frac {\pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i \, m + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i \, m + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, x e^{\left (m \log \left (x\right ) + m\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(pi*sqrt(b^2*d^2*n^2)*e^(m - m*log(c)/n - a*m/(b*n) - log(c)/n - a/(b*n) - 1/2*I*m^2/(pi*b^2*d^2*n^2) - I
*m/(pi*b^2*d^2*n^2) - 1/2*I/(pi*b^2*d^2*n^2))*fresnel_cos((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*
b*d^2*n + I*m + I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2*d^2*n^2)*e^(m - m*log(c)/n - a*m/(b*n) -
log(c)/n - a/(b*n) + 1/2*I*m^2/(pi*b^2*d^2*n^2) + I*m/(pi*b^2*d^2*n^2) + 1/2*I/(pi*b^2*d^2*n^2))*fresnel_cos((
pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - I*m - I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - I*
pi*sqrt(b^2*d^2*n^2)*e^(m - m*log(c)/n - a*m/(b*n) - log(c)/n - a/(b*n) - 1/2*I*m^2/(pi*b^2*d^2*n^2) - I*m/(pi
*b^2*d^2*n^2) - 1/2*I/(pi*b^2*d^2*n^2))*fresnel_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*
n + I*m + I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + I*pi*sqrt(b^2*d^2*n^2)*e^(m - m*log(c)/n - a*m/(b*n) - log(
c)/n - a/(b*n) + 1/2*I*m^2/(pi*b^2*d^2*n^2) + I*m/(pi*b^2*d^2*n^2) + 1/2*I/(pi*b^2*d^2*n^2))*fresnel_sin((pi*b
^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - I*m - I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 2*x*e^
(m*log(x) + m)*fresnel_cos(b*d*log(c*x^n) + a*d))/(m + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*x)**m*fresnelc(a*d + b*d*log(c*x**n)), 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((e*x)^m*fresnel_cos(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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