Integrand size = 8, antiderivative size = 90 \[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\frac {x^{1+m} \operatorname {CosIntegral}(b x)}{1+m}+\frac {i x^m (-i b x)^{-m} \Gamma (1+m,-i b x)}{2 b (1+m)}-\frac {i x^m (i b x)^{-m} \Gamma (1+m,i b x)}{2 b (1+m)} \]
x^(1+m)*Ci(b*x)/(1+m)+1/2*I*x^m*GAMMA(1+m,-I*b*x)/b/(1+m)/((-I*b*x)^m)-1/2 *I*x^m*GAMMA(1+m,I*b*x)/b/(1+m)/((I*b*x)^m)
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87 \[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\frac {x^m \left (2 x \operatorname {CosIntegral}(b x)+\frac {i \left (b^2 x^2\right )^{-m} \left ((i b x)^m \Gamma (1+m,-i b x)-(-i b x)^m \Gamma (1+m,i b x)\right )}{b}\right )}{2 (1+m)} \]
(x^m*(2*x*CosIntegral[b*x] + (I*((I*b*x)^m*Gamma[1 + m, (-I)*b*x] - ((-I)* b*x)^m*Gamma[1 + m, I*b*x]))/(b*(b^2*x^2)^m)))/(2*(1 + m))
Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7058, 27, 3042, 3788, 26, 2612}
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.
| \(\displaystyle \int x^m \operatorname {CosIntegral}(b x) \, dx\) |
| \(\Big \downarrow \) 7058 |
| \(\displaystyle \frac {x^{m+1} \operatorname {CosIntegral}(b x)}{m+1}-\frac {b \int \frac {x^m \cos (b x)}{b}dx}{m+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{m+1} \operatorname {CosIntegral}(b x)}{m+1}-\frac {\int x^m \cos (b x)dx}{m+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{m+1} \operatorname {CosIntegral}(b x)}{m+1}-\frac {\int x^m \sin \left (b x+\frac {\pi }{2}\right )dx}{m+1}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {x^{m+1} \operatorname {CosIntegral}(b x)}{m+1}-\frac {\frac {1}{2} i \int -i e^{-i b x} x^mdx-\frac {1}{2} i \int i e^{i b x} x^mdx}{m+1}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {x^{m+1} \operatorname {CosIntegral}(b x)}{m+1}-\frac {\frac {1}{2} \int e^{-i b x} x^mdx+\frac {1}{2} \int e^{i b x} x^mdx}{m+1}\) |
| \(\Big \downarrow \) 2612 |
| \(\displaystyle \frac {x^{m+1} \operatorname {CosIntegral}(b x)}{m+1}-\frac {\frac {i x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b}-\frac {i x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}}{m+1}\) |
(x^(1 + m)*CosIntegral[b*x])/(1 + m) - (((-1/2*I)*x^m*Gamma[1 + m, (-I)*b* x])/(b*((-I)*b*x)^m) + ((I/2)*x^m*Gamma[1 + m, I*b*x])/(b*(I*b*x)^m))/(1 + m)
3.1.69.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[CosIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] : > Simp[(c + d*x)^(m + 1)*(CosIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/(d *(m + 1)) Int[(c + d*x)^(m + 1)*(Cos[a + b*x]/(a + b*x)), x], x] /; FreeQ [{a, b, c, d, m}, x] && NeQ[m, -1]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.67 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.38
| method | result | size |
| meijerg | \(2^{m -1} b^{-m -1} \sqrt {\pi }\, \left (-\frac {2^{-m -1} x^{3+m} b^{3+m} \operatorname {hypergeom}\left (\left [1, 1, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, 2, 2, \frac {5}{2}+\frac {m}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{\sqrt {\pi }\, \left (3+m \right )}+\frac {2 \left (\Psi \left (\frac {1}{2}+\frac {m}{2}\right )+2 \gamma -\Psi \left (\frac {3}{2}+\frac {m}{2}\right )+2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x^{1+m} 2^{-m -1} b^{1+m}}{\sqrt {\pi }\, \left (1+m \right )}\right )\) | \(124\) |
2^(m-1)*b^(-m-1)*Pi^(1/2)*(-2^(-m-1)/Pi^(1/2)/(3+m)*x^(3+m)*b^(3+m)*hyperg eom([1,1,3/2+1/2*m],[3/2,2,2,5/2+1/2*m],-1/4*b^2*x^2)+2*(Psi(1/2+1/2*m)+2* gamma-Psi(3/2+1/2*m)+2*ln(x)+2*ln(b))/Pi^(1/2)*x^(1+m)*2^(-m-1)*b^(1+m)/(1 +m))
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.21 \[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\frac {2 \, \pi b x x^{m} \operatorname {C}\left (b x\right ) - i \, {\left (\cosh \left (\frac {1}{2} \, m \log \left (\frac {1}{2} i \, \pi b^{2}\right )\right ) - \sinh \left (\frac {1}{2} \, m \log \left (\frac {1}{2} i \, \pi b^{2}\right )\right )\right )} \Gamma \left (\frac {1}{2} \, m + 1, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + i \, {\left (\cosh \left (\frac {1}{2} \, m \log \left (-\frac {1}{2} i \, \pi b^{2}\right )\right ) - \sinh \left (\frac {1}{2} \, m \log \left (-\frac {1}{2} i \, \pi b^{2}\right )\right )\right )} \Gamma \left (\frac {1}{2} \, m + 1, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )}{2 \, \pi {\left (b m + b\right )}} \]
1/2*(2*pi*b*x*x^m*fresnel_cos(b*x) - I*(cosh(1/2*m*log(1/2*I*pi*b^2)) - si nh(1/2*m*log(1/2*I*pi*b^2)))*gamma(1/2*m + 1, 1/2*I*pi*b^2*x^2) + I*(cosh( 1/2*m*log(-1/2*I*pi*b^2)) - sinh(1/2*m*log(-1/2*I*pi*b^2)))*gamma(1/2*m + 1, -1/2*I*pi*b^2*x^2))/(pi*(b*m + b))
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (70) = 140\).
Time = 1.01 (sec) , antiderivative size = 700, normalized size of antiderivative = 7.78 \[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\frac {4 \cdot 2^{m} b b^{- m - 1} m x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \log {\left (b^{2} x^{2} \right )} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {8 \cdot 2^{m} \gamma b b^{- m - 1} m x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {4 \cdot 2^{m} b b^{- m - 1} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \log {\left (b^{2} x^{2} \right )} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {8 \cdot 2^{m} b b^{- m - 1} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {8 \cdot 2^{m} \gamma b b^{- m - 1} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {b^{- m - 1} b^{m + 3} m^{2} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {2 b^{- m - 1} b^{m + 3} m x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {b^{- m - 1} b^{m + 3} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \]
4*2**m*b*b**(-m - 1)*m*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*log( b**2*x**2)*gamma(m/2 + 5/2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/ 2) + 8*gamma(m/2 + 5/2)) + 8*2**m*EulerGamma*b*b**(-m - 1)*m*x*sqrt(exp(-2 *m*log(2))*exp(m*log(b**2*x**2)))*gamma(m/2 + 5/2)/(8*m**2*gamma(m/2 + 5/2 ) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) + 4*2**m*b*b**(-m - 1)*x*s qrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*log(b**2*x**2)*gamma(m/2 + 5/2 )/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - 8*2**m*b*b**(-m - 1)*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*gamma (m/2 + 5/2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) + 8*2**m*EulerGamma*b*b**(-m - 1)*x*sqrt(exp(-2*m*log(2))*exp(m*l og(b**2*x**2)))*gamma(m/2 + 5/2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - b**(-m - 1)*b**(m + 3)*m**2*x**(m + 3)*gam ma(m/2 + 3/2)*hyper((1, 1, m/2 + 3/2), (3/2, 2, 2, m/2 + 5/2), -b**2*x**2/ 4)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - 2*b**(-m - 1)*b**(m + 3)*m*x**(m + 3)*gamma(m/2 + 3/2)*hyper((1, 1, m/2 + 3/2), (3/2, 2, 2, m/2 + 5/2), -b**2*x**2/4)/(8*m**2*gamma(m/2 + 5/2) + 1 6*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - b**(-m - 1)*b**(m + 3)*x**(m + 3)*gamma(m/2 + 3/2)*hyper((1, 1, m/2 + 3/2), (3/2, 2, 2, m/2 + 5/2), -b* *2*x**2/4)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2))
\[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\int { x^{m} \operatorname {C}\left (b x\right ) \,d x } \]
\[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\int { x^{m} \operatorname {C}\left (b x\right ) \,d x } \]
Timed out. \[ \int x^m \operatorname {CosIntegral}(b x) \, dx=\int x^m\,\mathrm {cosint}\left (b\,x\right ) \,d x \]