3.1.69 \(\int x^m \text {CosIntegral}(b x) \, dx\) [69]

Optimal. Leaf size=90 \[ \frac {x^{1+m} \text {CosIntegral}(b x)}{1+m}+\frac {i x^m (-i b x)^{-m} \Gamma (1+m,-i b x)}{2 b (1+m)}-\frac {i x^m (i b x)^{-m} \Gamma (1+m,i b x)}{2 b (1+m)} \]

[Out]

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)

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Rubi [A]
time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3388, 2212} \begin {gather*} \frac {i x^m (-i b x)^{-m} \text {Gamma}(m+1,-i b x)}{2 b (m+1)}-\frac {i x^m (i b x)^{-m} \text {Gamma}(m+1,i b x)}{2 b (m+1)}+\frac {x^{m+1} \text {CosIntegral}(b x)}{m+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*CosIntegral[b*x],x]

[Out]

(x^(1 + m)*CosIntegral[b*x])/(1 + m) + ((I/2)*x^m*Gamma[1 + m, (-I)*b*x])/(b*(1 + m)*((-I)*b*x)^m) - ((I/2)*x^
m*Gamma[1 + m, I*b*x])/(b*(1 + m)*(I*b*x)^m)

Rule 12

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

Rule 2212

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]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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]

Rule 6639

Int[CosIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(CosIntegr
al[a + b*x]/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(Cos[a + b*x]/(a + b*x)), x], x] /; F
reeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^m \text {Ci}(b x) \, dx &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}-\frac {b \int \frac {x^m \cos (b x)}{b} \, dx}{1+m}\\ &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}-\frac {\int x^m \cos (b x) \, dx}{1+m}\\ &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}-\frac {\int e^{-i b x} x^m \, dx}{2 (1+m)}-\frac {\int e^{i b x} x^m \, dx}{2 (1+m)}\\ &=\frac {x^{1+m} \text {Ci}(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)}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 78, normalized size = 0.87 \begin {gather*} \frac {x^m \left (2 x \text {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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*CosIntegral[b*x],x]

[Out]

(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))

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.55, size = 124, normalized size = 1.38

method	result	size
meijerg	\(2^{-1+m} b^{-1-m} \sqrt {\pi }\, \left (-\frac {2^{-1-m} x^{3+m} b^{3+m} \hypergeom \left (\left [1, 1, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, 2, 2, \frac {5}{2}+\frac {m}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{\sqrt {\pi }\, \left (3+m \right )}+\frac {2 \left (\Psi \left (\frac {1}{2}+\frac {m}{2}\right )+2 \gamma -\Psi \left (\frac {3}{2}+\frac {m}{2}\right )+2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x^{1+m} 2^{-1-m} b^{1+m}}{\sqrt {\pi }\, \left (1+m \right )}\right )\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*Ci(b*x),x,method=_RETURNVERBOSE)

[Out]

2^(-1+m)*b^(-1-m)*Pi^(1/2)*(-2^(-1-m)/Pi^(1/2)/(3+m)*x^(3+m)*b^(3+m)*hypergeom([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^(-1-m)*b^(1+m)
/(1+m))

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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(x^m*fresnel_cos(b*x),x, algorithm="maxima")

[Out]

integrate(x^m*fresnel_cos(b*x), x)

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Fricas [A]
time = 0.11, size = 109, normalized size = 1.21 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*fresnel_cos(b*x),x, algorithm="fricas")

[Out]

1/2*(2*pi*b*x*x^m*fresnel_cos(b*x) - 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) + 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))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (70) = 140\).
time = 0.98, size = 654, normalized size = 7.27 \begin {gather*} \frac {4 \cdot 2^{m} b^{- m} 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^{- m} 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^{- 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} b^{- 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 {8 \cdot 2^{m} \gamma b^{- 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 {b^{2} m^{2} x^{3} x^{m} \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^{2} m x^{3} x^{m} \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^{2} x^{3} x^{m} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*Ci(b*x),x)

[Out]

4*2**m*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)/(b**m*(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*m*x*sqrt(exp(-2*m*log(2))*exp(m*lo
g(b**2*x**2)))*gamma(m/2 + 5/2)/(b**m*(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)))
+ 4*2**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)/(b**m*(8*m**2*gamma(m/
2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2))) - 8*2**m*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2))
)*gamma(m/2 + 5/2)/(b**m*(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2))) + 8*2**m*Eule
rGamma*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*gamma(m/2 + 5/2)/(b**m*(8*m**2*gamma(m/2 + 5/2) + 16*m*g
amma(m/2 + 5/2) + 8*gamma(m/2 + 5/2))) - b**2*m**2*x**3*x**m*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)) - 2*b**2
*m*x**3*x**m*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)) - b**2*x**3*x**m*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)
)

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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(x^m*fresnel_cos(b*x),x, algorithm="giac")

[Out]

integrate(x^m*fresnel_cos(b*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,\mathrm {cosint}\left (b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cosint(b*x),x)

[Out]

int(x^m*cosint(b*x), x)

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