3.69 \(\int x^m \text {Ci}(b x) \, dx\)

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

[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 = 0.08, 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 = {6504, 12, 3307, 2181} \[ \frac {i x^m (-i b x)^{-m} \text {Gamma}(m+1,-i b x)}{2 b (m+1)}-\frac {i x^m (i b x)^{-m} \text {Gamma}(m+1,i b x)}{2 b (m+1)}+\frac {x^{m+1} \text {CosIntegral}(b x)}{m+1} \]

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 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3307

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 6504

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 = 0.11, size = 78, normalized size = 0.87 \[ \frac {x^m \left (2 x \text {Ci}(b x)+\frac {i \left (b^2 x^2\right )^{-m} \left ((i b x)^m \Gamma (m+1,-i b x)-(-i b x)^m \Gamma (m+1,i b x)\right )}{b}\right )}{2 (m+1)} \]

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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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {Ci}\left (b x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*cos_integral(b*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} {\rm Ci}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*Ci(b*x), x)

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maple [C]  time = 0.08, size = 124, normalized size = 1.38 \[ 2^{m -1} b^{-m -1} \sqrt {\pi }\, \left (-\frac {2^{-m -1} x^{m +3} b^{m +3} \hypergeom \left (\left [1, 1, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, 2, 2, \frac {5}{2}+\frac {m}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{\sqrt {\pi }\, \left (m +3\right )}+\frac {2 \left (\Psi \left (\frac {1}{2}+\frac {m}{2}\right )+2 \gamma -\Psi \left (\frac {3}{2}+\frac {m}{2}\right )+2 \ln \relax (x )+2 \ln \relax (b )\right ) x^{1+m} 2^{-m -1} b^{1+m}}{\sqrt {\pi }\, \left (1+m \right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} {\rm Ci}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*Ci(b*x), x)

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

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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sympy [B]  time = 1.43, size = 654, normalized size = 7.27 \[ \frac {4 \cdot 2^{m} b^{- m} m x \sqrt {e^{- 2 m \log {\relax (2 )}} 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 {\relax (2 )}} 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 {\relax (2 )}} 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 {\relax (2 )}} 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 {\relax (2 )}} 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 )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*2**m*b**(-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)/(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**(-m)*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**(-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*gamm
a(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - 8*2**m*b**(-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)) + 8*2**m*
EulerGamma*b**(-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)) - 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*gamm
a(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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