3.60 \(\int (e x)^m S(d (a+b \log (c x^n))) \, dx\)

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

[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)*FresnelS(d*(a+b*ln(c*x^n)))/e/(1+m)

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Rubi [A]  time = 0.70, antiderivative size = 280, normalized size of antiderivative = 1.00, 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 = {6471, 4617, 2278, 2274, 15, 20, 2276, 2234, 2204, 2205} \[ \frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (\frac {i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{m+1}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (-\frac {i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2 n-i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{m+1}+\frac {(e x)^{m+1} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*FresnelS[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
esnelS[d*(a + b*Log[c*x^n])])/(e*(1 + m))

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 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]

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 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 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 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 4617

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)], x_Symbol] :> Dist[I/2, Int[(e*x)^m/
E^(I*d*(a + b*Log[c*x^n])^2), x], x] - Dist[I/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 6471

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

Rubi steps

\begin {align*} \int (e x)^m S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {(e x)^{1+m} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int (e x)^m \sin \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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(i 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 {(i 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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(i 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 {(i 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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(i 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 {(i 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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i 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 (i 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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i 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 (i 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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i 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 (i 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} S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i 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 (i 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} S\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 = 6.22, size = 244, normalized size = 0.87 \[ \frac {(e x)^m \left (4 x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\sqrt [4]{-1} \sqrt {2} x^{-m} \exp \left (-\frac {(m+1) \left (2 \pi a b d^2 n+2 \pi b^2 d^2 n \left (\log \left (c x^n\right )-n \log (x)\right )+i m+i\right )}{2 \pi b^2 d^2 n^2}\right ) \left (\text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+i m+i\right )}{\sqrt {\pi } b d n}\right )+e^{\frac {i (m+1)^2}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {(-1)^{3/4} \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {2 \pi } b d n}\right )\right )\right )}{4 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^m*(-(((-1)^(1/4)*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*S
qrt[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*FresnelS[d*(a + b*Log[c*x^n])]))/(4*(1 + m))

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} {\rm fresnels}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} {\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \mathrm {S}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} {\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*x)**m*fresnels(a*d + b*d*log(c*x**n)), x)

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