3.1.24 \(\int \frac {\sin (c+d x)}{x^2 (a+b x)} \, dx\) [24]

Optimal. Leaf size=114 \[ \frac {d \cos (c) \text {Ci}(d x)}{a}-\frac {b \text {Ci}(d x) \sin (c)}{a^2}+\frac {b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^2}-\frac {\sin (c+d x)}{a x}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a}+\frac {b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2} \]

[Out]

d*Ci(d*x)*cos(c)/a-b*cos(c)*Si(d*x)/a^2+b*cos(-c+a*d/b)*Si(a*d/b+d*x)/a^2-b*Ci(d*x)*sin(c)/a^2-d*Si(d*x)*sin(c
)/a-b*Ci(a*d/b+d*x)*sin(-c+a*d/b)/a^2-sin(d*x+c)/a/x

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Rubi [A]
time = 0.24, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3380, 3383} \begin {gather*} -\frac {b \sin (c) \text {CosIntegral}(d x)}{a^2}+\frac {b \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}+\frac {b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {d \cos (c) \text {CosIntegral}(d x)}{a}-\frac {d \sin (c) \text {Si}(d x)}{a}-\frac {\sin (c+d x)}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^2*(a + b*x)),x]

[Out]

(d*Cos[c]*CosIntegral[d*x])/a - (b*CosIntegral[d*x]*Sin[c])/a^2 + (b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/
b])/a^2 - Sin[c + d*x]/(a*x) - (b*Cos[c]*SinIntegral[d*x])/a^2 - (d*Sin[c]*SinIntegral[d*x])/a + (b*Cos[c - (a
*d)/b]*SinIntegral[(a*d)/b + d*x])/a^2

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3380

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

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 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]
time = 0.25, size = 101, normalized size = 0.89 \begin {gather*} \frac {x \text {Ci}(d x) (a d \cos (c)-b \sin (c))+b x \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )-a \sin (c+d x)-b x \cos (c) \text {Si}(d x)-a d x \sin (c) \text {Si}(d x)+b x \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{a^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(x^2*(a + b*x)),x]

[Out]

(x*CosIntegral[d*x]*(a*d*Cos[c] - b*Sin[c]) + b*x*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] - a*Sin[c + d*x] -
b*x*Cos[c]*SinIntegral[d*x] - a*d*x*Sin[c]*SinIntegral[d*x] + b*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/
(a^2*x)

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Maple [A]
time = 0.10, size = 144, normalized size = 1.26

method	result	size
derivativedivides	\(d \left (\frac {-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )}{a}-\frac {b \left (\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )\right )}{a^{2} d}+\frac {b^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a^{2} d}\right )\)	\(144\)
default	\(d \left (\frac {-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )}{a}-\frac {b \left (\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )\right )}{a^{2} d}+\frac {b^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a^{2} d}\right )\)	\(144\)
risch	\(-\frac {d \,{\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a}+\frac {i b \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{2}}-\frac {i b \,{\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}-\frac {i b \,{\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a^{2}}-\frac {d \,{\mathrm e}^{-i c} \expIntegral \left (1, i d x \right )}{2 a}+\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, i d x \right ) b}{2 a^{2}}-\frac {\sin \left (d x +c \right )}{a x}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x^2/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

d*(1/a*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)*cos(c))-1/a^2*b/d*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+b^2/a^2/d*(Si
(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b))

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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(sin(d*x+c)/x^2/(b*x+a),x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x + a)*x^2), x)

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Fricas [A]
time = 0.35, size = 157, normalized size = 1.38 \begin {gather*} \frac {2 \, b x \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + {\left (a d x \operatorname {Ci}\left (d x\right ) + a d x \operatorname {Ci}\left (-d x\right ) - 2 \, b x \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) - 2 \, a \sin \left (d x + c\right ) - {\left (2 \, a d x \operatorname {Si}\left (d x\right ) + b x \operatorname {Ci}\left (d x\right ) + b x \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right ) - {\left (b x \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + b x \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, a^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^2/(b*x+a),x, algorithm="fricas")

[Out]

1/2*(2*b*x*cos(-(b*c - a*d)/b)*sin_integral((b*d*x + a*d)/b) + (a*d*x*cos_integral(d*x) + a*d*x*cos_integral(-
d*x) - 2*b*x*sin_integral(d*x))*cos(c) - 2*a*sin(d*x + c) - (2*a*d*x*sin_integral(d*x) + b*x*cos_integral(d*x)
+ b*x*cos_integral(-d*x))*sin(c) - (b*x*cos_integral((b*d*x + a*d)/b) + b*x*cos_integral(-(b*d*x + a*d)/b))*s
in(-(b*c - a*d)/b))/(a^2*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x**2/(b*x+a),x)

[Out]

Integral(sin(c + d*x)/(x**2*(a + b*x)), x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.83, size = 2897, normalized size = 25.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^2/(b*x+a),x, algorithm="giac")

[Out]

-1/2*(a*d*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*d*x*real_part(cos_in
tegral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)
^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*d*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b
)^2 + 4*a*d*x*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - b*x*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 + b*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^
2 + b*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b*x*sin_integral(d*x)*t
an(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + a*d*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*d*x*real_part(cos_integr
al(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*b*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2
*tan(1/2*a*d/b) - 2*b*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - a*d
*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a*d*x*real_part(cos_integral(-d*x))*tan(1/2*
d*x)^2*tan(1/2*a*d/b)^2 + 2*b*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^
2 + 2*b*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b*x*real_part(cos_integr
al(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b*x*real_part(cos_integral(-d*x))*tan(1/2*d*x
)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + a*d*x*real_part(cos_integral(d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*d*x*rea
l_part(cos_integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2
*tan(1/2*c) - 2*a*d*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*d*x*sin_integral(d*x)*tan(
1/2*d*x)^2*tan(1/2*c) + b*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - b*x*imag_part(c
os_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - b*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1
/2*c)^2 + b*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*b*x*sin_integral(d*x)*tan(1/2*d*x)
^2*tan(1/2*c)^2 + 2*b*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*b*x*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*b*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/
2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 8*b*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d
/b) + b*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b*x*imag_part(cos_integral(d*
x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - b*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^
2 - b*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*b*x*sin_integral(d*x)*tan(1/2*d*x)^2
*tan(1/2*a*d/b)^2 + 2*b*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a*d*x*imag_part(co
s_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*d*x*imag_part(cos_integral(-d*x))*tan(1/2*c)*tan(1/2*a*d/b)
^2 + 4*a*d*x*sin_integral(d*x)*tan(1/2*c)*tan(1/2*a*d/b)^2 - b*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
c)^2*tan(1/2*a*d/b)^2 - b*x*imag_part(cos_integral(d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b*x*imag_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b*x*imag_part(cos_integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*
d/b)^2 - 2*b*x*sin_integral(d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 - a*d*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2 - a*d*x*real_part(cos_integral(-d*x))
*tan(1/2*d*x)^2 - 2*b*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*b*x*real_part(cos_i
ntegral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*b*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c
) + 2*b*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + a*d*x*real_part(cos_integral(d*x))*tan(1/2
*c)^2 + a*d*x*real_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*b*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x)^2*tan(1/2*a*d/b) + 2*b*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*b*x*real
_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*b*x*real_part(cos_integral(-d*x - a*d/b))*tan
(1/2*c)^2*tan(1/2*a*d/b) - a*d*x*real_part(cos_integral(d*x))*tan(1/2*a*d/b)^2 - a*d*x*real_part(cos_integral(
-d*x))*tan(1/2*a*d/b)^2 + 2*b*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b*x*real_
part(cos_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*t
an(1/2*a*d/b)^2 + 2*b*x*real_part(cos_integral(-d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a*tan(1/2*d*x)^2*tan(1/2
*c)*tan(1/2*a*d/b)^2 - 4*a*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b*x*imag_part(cos_integral(d*x + a*d/b
))*tan(1/2*d*x)^2 + b*x*imag_part(cos_integral(...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{x^2\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x^2*(a + b*x)),x)

[Out]

int(sin(c + d*x)/(x^2*(a + b*x)), x)

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