3.2.5 \(\int \frac {\text {CosIntegral}(d (a+b \log (c x^n)))}{x^3} \, dx\) [105]

Optimal. Leaf size=135 \[ -\frac {\text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \]

[Out]

-1/2*Ci(d*(a+b*ln(c*x^n)))/x^2+1/4*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(2-I*b*d*n)*(a+b*ln(c*x^n))/b/n)/x^2+1/4*exp
(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(2+I*b*d*n)*(a+b*ln(c*x^n))/b/n)/x^2

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Rubi [A]
time = 0.17, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6662, 12, 4586, 2347, 2209} \begin {gather*} -\frac {\text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(i b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[CosIntegral[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

-1/2*CosIntegral[d*(a + b*Log[c*x^n])]/x^2 + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - I*b*d*n)*(a
+ b*Log[c*x^n]))/(b*n))])/(4*x^2) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + I*b*d*n)*(a + b*Log[c
*x^n]))/(b*n))])/(4*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2347

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

Rule 4586

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.
)*(x_))^(r_.), x_Symbol] :> Dist[((i*x)^r*(1/((c*x^n)^(I*b*d)*(2*x^(r - I*b*d*n)))))/E^(I*a*d), Int[x^(r - I*b
*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] + Dist[E^(I*a*d)*(i*x)^r*((c*x^n)^(I*b*d)/(2*x^(r + I*b*d*n))), Int[x^(
r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]

Rule 6662

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

Rubi steps

\begin {align*} \int \frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b d n) \int \frac {\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx\\ &=-\frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b n) \int \frac {\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx\\ &=-\frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} \left (b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{-3-i b d n}}{a+b \log \left (c x^n\right )} \, dx+\frac {1}{4} \left (b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac {x^{-3+i b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=-\frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (b e^{-i a d} \left (c x^n\right )^{-i b d-\frac {-2-i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-2-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2}+\frac {\left (b e^{i a d} \left (c x^n\right )^{i b d-\frac {-2+i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-2+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2}\\ &=-\frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}\\ \end {align*}

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Mathematica [A]
time = 1.00, size = 105, normalized size = 0.78 \begin {gather*} \frac {-2 \text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (\text {Ei}\left (-\frac {i (-2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text {Ei}\left (\frac {i (2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[CosIntegral[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

(-2*CosIntegral[d*(a + b*Log[c*x^n])] + E^((2*a)/(b*n))*(c*x^n)^(2/n)*(ExpIntegralEi[((-I)*(-2*I + b*d*n)*(a +
b*Log[c*x^n]))/(b*n)] + ExpIntegralEi[(I*(2*I + b*d*n)*(a + b*Log[c*x^n]))/(b*n)]))/(4*x^2)

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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\cosineIntegral \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(d*(a+b*ln(c*x^n)))/x^3,x)

[Out]

int(Ci(d*(a+b*ln(c*x^n)))/x^3,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(fresnel_cos(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")

[Out]

integrate(fresnel_cos((b*log(c*x^n) + a)*d)/x^3, 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. 460 vs. \(2 (121) = 242\).
time = 0.38, size = 460, normalized size = 3.41 \begin {gather*} \frac {\pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\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 + 2 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}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\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 - 2 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}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\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 + 2 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}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\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 - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")

[Out]

1/4*(pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) + 2*I/(pi*b^2*d^2*n^2))*fresnel_cos((pi*b^2*d^2*n^2*lo
g(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n + 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2*d^2*n^2)*x^
2*e^(2*log(c)/n + 2*a/(b*n) - 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 - 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + I*pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n
) + 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 + 2*I)*sqrt(
b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - I*pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) - 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 - 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n
^2)) - 2*fresnel_cos(b*d*log(c*x^n) + a*d))/x^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(d*(a+b*ln(c*x**n)))/x**3,x)

[Out]

Integral(Ci(a*d + b*d*log(c*x**n))/x**3, 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^3,x, algorithm="giac")

[Out]

integrate(fresnel_cos((b*log(c*x^n) + a)*d)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosint(d*(a + b*log(c*x^n)))/x^3,x)

[Out]

int(cosint(d*(a + b*log(c*x^n)))/x^3, x)

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