Integrand size = 19, antiderivative size = 172 \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)} \]
(e*x)^(1+m)*Ci(d*(a+b*ln(c*x^n)))/e/(1+m)-1/2*x*(e*x)^m*Ei((1+m-I*b*d*n)*( a+b*ln(c*x^n))/b/n)/exp(a*(1+m)/b/n)/(1+m)/((c*x^n)^((1+m)/n))-1/2*x*(e*x) ^m*Ei((1+m+I*b*d*n)*(a+b*ln(c*x^n))/b/n)/exp(a*(1+m)/b/n)/(1+m)/((c*x^n)^( (1+m)/n))
Time = 2.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.72 \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (2 x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \left (\operatorname {ExpIntegralEi}\left (\frac {(1+m-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+m+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right )}{2 (1+m)} \]
((e*x)^m*(2*x*CosIntegral[d*(a + b*Log[c*x^n])] - (ExpIntegralEi[((1 + m - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)] + ExpIntegralEi[((1 + m + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^(((1 + m)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b *n))*x^m)))/(2*(1 + m))
Time = 0.66 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {7081, 27, 5001, 2747, 2609}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7081 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \int \frac {(e x)^m \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )}dx}{m+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \int \frac {(e x)^m \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )}dx}{m+1}\) |
| \(\Big \downarrow \) 5001 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \left (\frac {1}{2} e^{-i a d} (e x)^m \left (c x^n\right )^{-i b d} x^{-m+i b d n} \int \frac {x^{m-i b d n}}{a+b \log \left (c x^n\right )}dx+\frac {1}{2} e^{i a d} (e x)^m \left (c x^n\right )^{i b d} x^{-m-i b d n} \int \frac {x^{m+i b d n}}{a+b \log \left (c x^n\right )}dx\right )}{m+1}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \left (\frac {x e^{-i a d} (e x)^m \left (c x^n\right )^{-\frac {-i b d n+m+1}{n}-i b d} \int \frac {\left (c x^n\right )^{\frac {m-i b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}+\frac {x e^{i a d} (e x)^m \left (c x^n\right )^{i b d-\frac {i b d n+m+1}{n}} \int \frac {\left (c x^n\right )^{\frac {m+i b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}\right )}{m+1}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \left (\frac {x (e x)^m e^{-\frac {a (-i b d n+m+1)}{b n}-i a d} \left (c x^n\right )^{-\frac {-i b d n+m+1}{n}-i b d} \operatorname {ExpIntegralEi}\left (\frac {(m-i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}+\frac {x (e x)^m e^{i a d-\frac {a (i b d n+m+1)}{b n}} \left (c x^n\right )^{i b d-\frac {i b d n+m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}\right )}{m+1}\) |
((e*x)^(1 + m)*CosIntegral[d*(a + b*Log[c*x^n])])/(e*(1 + m)) - (b*n*((E^( (-I)*a*d - (a*(1 + m - I*b*d*n))/(b*n))*x*(e*x)^m*(c*x^n)^((-I)*b*d - (1 + m - I*b*d*n)/n)*ExpIntegralEi[((1 + m - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n )])/(2*b*n) + (E^(I*a*d - (a*(1 + m + I*b*d*n))/(b*n))*x*(e*x)^m*(c*x^n)^( I*b*d - (1 + m + I*b*d*n)/n)*ExpIntegralEi[((1 + m + I*b*d*n)*(a + b*Log[c *x^n]))/(b*n)])/(2*b*n)))/(1 + m)
3.2.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(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]
Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*(x _)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.), x_Symbol] :> Simp[((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] + Simp[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]
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] - Simp[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] && N eQ[m, -1]
\[\int \left (e x \right )^{m} \operatorname {Ci}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (164) = 328\).
Time = 0.29 (sec) , antiderivative size = 689, normalized size of antiderivative = 4.01 \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {\pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \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 \log \left (e\right ) - \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 \log \left (e\right ) - \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 \log \left (e\right ) - \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 (e\right ) + m \log \left (x\right )\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, {\left (m + 1\right )}} \]
-1/2*(pi*sqrt(b^2*d^2*n^2)*e^(m*log(e) - 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*log(e) - 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))*fre snel_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*lo g(e) - 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*log(e) - 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(e) + m*log(x))*fresnel_cos(b*d*log(c*x^n) + a*d)) /(m + 1)
\[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]