3.2.10 \(\int \text {CosIntegral}(b x) \sin (b x) \, dx\) [110]

Optimal. Leaf size=35 \[ -\frac {\cos (b x) \text {CosIntegral}(b x)}{b}+\frac {\text {CosIntegral}(2 b x)}{2 b}+\frac {\log (x)}{2 b} \]

[Out]

1/2*Ci(2*b*x)/b-Ci(b*x)*cos(b*x)/b+1/2*ln(x)/b

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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6653, 12, 3393, 3383} \begin {gather*} \frac {\text {CosIntegral}(2 b x)}{2 b}-\frac {\text {CosIntegral}(b x) \cos (b x)}{b}+\frac {\log (x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[CosIntegral[b*x]*Sin[b*x],x]

[Out]

-((Cos[b*x]*CosIntegral[b*x])/b) + CosIntegral[2*b*x]/(2*b) + Log[x]/(2*b)

Rule 12

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

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 6653

Int[CosIntegral[(c_.) + (d_.)*(x_)]*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(-Cos[a + b*x])*(CosIntegral[c
+ d*x]/b), x] + Dist[d/b, Int[Cos[a + b*x]*(Cos[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]

Rubi steps

\begin {align*} \int \text {Ci}(b x) \sin (b x) \, dx &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\int \frac {\cos ^2(b x)}{b x} \, dx\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\int \frac {\cos ^2(b x)}{x} \, dx}{b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\log (x)}{2 b}+\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\text {Ci}(2 b x)}{2 b}+\frac {\log (x)}{2 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.06 \begin {gather*} -\frac {\cos (b x) \text {CosIntegral}(b x)}{b}+\frac {\text {CosIntegral}(2 b x)}{2 b}+\frac {\log (b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[CosIntegral[b*x]*Sin[b*x],x]

[Out]

-((Cos[b*x]*CosIntegral[b*x])/b) + CosIntegral[2*b*x]/(2*b) + Log[b*x]/(2*b)

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Maple [A]
time = 0.35, size = 29, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {-\cosineIntegral \left (b x \right ) \cos \left (b x \right )+\frac {\ln \left (b x \right )}{2}+\frac {\cosineIntegral \left (2 b x \right )}{2}}{b}\)	\(29\)
default	\(\frac {-\cosineIntegral \left (b x \right ) \cos \left (b x \right )+\frac {\ln \left (b x \right )}{2}+\frac {\cosineIntegral \left (2 b x \right )}{2}}{b}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x)*sin(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b*(-Ci(b*x)*cos(b*x)+1/2*ln(b*x)+1/2*Ci(2*b*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(b*x)*sin(b*x),x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)*sin(b*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (31) = 62\).
time = 0.37, size = 145, normalized size = 4.14 \begin {gather*} -\frac {2 \, b \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right ) - \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right )}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*cos(b*x)*fresnel_cos(b*x) - sqrt(b^2)*cos(1/2/pi)*fresnel_cos((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - sqrt(
b^2)*cos(1/2/pi)*fresnel_cos((pi*b*x - 1)*sqrt(b^2)/(pi*b)) - sqrt(b^2)*fresnel_sin((pi*b*x + 1)*sqrt(b^2)/(pi
*b))*sin(1/2/pi) - sqrt(b^2)*fresnel_sin((pi*b*x - 1)*sqrt(b^2)/(pi*b))*sin(1/2/pi))/b^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(b*x)*Ci(b*x), 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(b*x)*sin(b*x),x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)*sin(b*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \frac {\ln \left (x\right )}{2\,b}+\frac {\mathrm {cosint}\left (2\,b\,x\right )}{2\,b}-\frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosint(b*x)*sin(b*x),x)

[Out]

log(x)/(2*b) + cosint(2*b*x)/(2*b) - (cosint(b*x)*cos(b*x))/b

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