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

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

[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))

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Rubi [A]  time = 0.607006, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {6472, 4618, 2278, 2274, 15, 20, 2276, 2234, 2204, 2205} \[ \frac{\left (\frac{1}{4}+\frac{i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (\frac{i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt{\pi } b d n}\right )}{m+1}-\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (-\frac{i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2 n-i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt{\pi } b d n}\right )}{m+1}+\frac{(e x)^{m+1} \text{FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]

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 6472

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*(d*(a + b*Log[c*x
^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rule 4618

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 2278

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

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 2276

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

Rule 2234

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 2204

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 2205

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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 5.35204, size = 244, normalized size = 0.87 \[ \frac{(e x)^m \left (4 x \text{FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+(-1)^{3/4} \sqrt{2} x^{-m} \exp \left (-\frac{(m+1) \left (2 \pi a b d^2 n+2 \pi b^2 d^2 n \left (\log \left (c x^n\right )-n \log (x)\right )+i m+i\right )}{2 \pi b^2 d^2 n^2}\right ) \left (\text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+i m+i\right )}{\sqrt{\pi } b d n}\right )-e^{\frac{i (m+1)^2}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{(-1)^{3/4} \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt{2 \pi } b d n}\right )\right )\right )}{4 (m+1)} \]

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 = 0.205, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\it FresnelC} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m}{\rm fresnelc}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x)^m*fresnelc(b*d*log(c*x^n) + a*d), x)

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

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., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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