3.37 \(\int \frac{\sin (c+d x)}{x (a+b x)^3} \, dx\)

Optimal. Leaf size=261 \[ -\frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^2 b}+\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^2 b}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^3}+\frac{\sin (c+d x)}{a^2 (a+b x)}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{2 a b^2}+\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{2 a b^2}+\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{d \cos (c+d x)}{2 a b (a+b x)} \]

[Out]

(d*Cos[c + d*x])/(2*a*b*(a + b*x)) - (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/(a^2*b) + (CosIntegral[d*
x]*Sin[c])/a^3 - (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 + (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (
a*d)/b])/(2*a*b^2) + Sin[c + d*x]/(2*a*(a + b*x)^2) + Sin[c + d*x]/(a^2*(a + b*x)) + (Cos[c]*SinIntegral[d*x])
/a^3 - (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3 + (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(
2*a*b^2) + (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(a^2*b)

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Rubi [A]  time = 0.542232, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3303, 3299, 3302, 3297} \[ -\frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^2 b}+\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^2 b}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^3}+\frac{\sin (c+d x)}{a^2 (a+b x)}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{2 a b^2}+\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{2 a b^2}+\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{d \cos (c+d x)}{2 a b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x)^3),x]

[Out]

(d*Cos[c + d*x])/(2*a*b*(a + b*x)) - (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/(a^2*b) + (CosIntegral[d*
x]*Sin[c])/a^3 - (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 + (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (
a*d)/b])/(2*a*b^2) + Sin[c + d*x]/(2*a*(a + b*x)^2) + Sin[c + d*x]/(a^2*(a + b*x)) + (Cos[c]*SinIntegral[d*x])
/a^3 - (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3 + (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(
2*a*b^2) + (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(a^2*b)

Rule 6742

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

Rule 3303

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 3299

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 3302

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 3297

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]

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{x (a+b x)^3} \, dx &=\int \left (\frac{\sin (c+d x)}{a^3 x}-\frac{b \sin (c+d x)}{a (a+b x)^3}-\frac{b \sin (c+d x)}{a^2 (a+b x)^2}-\frac{b \sin (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x} \, dx}{a^3}-\frac{b \int \frac{\sin (c+d x)}{a+b x} \, dx}{a^3}-\frac{b \int \frac{\sin (c+d x)}{(a+b x)^2} \, dx}{a^2}-\frac{b \int \frac{\sin (c+d x)}{(a+b x)^3} \, dx}{a}\\ &=\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{\sin (c+d x)}{a^2 (a+b x)}-\frac{d \int \frac{\cos (c+d x)}{a+b x} \, dx}{a^2}-\frac{d \int \frac{\cos (c+d x)}{(a+b x)^2} \, dx}{2 a}+\frac{\cos (c) \int \frac{\sin (d x)}{x} \, dx}{a^3}-\frac{\left (b \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^3}+\frac{\sin (c) \int \frac{\cos (d x)}{x} \, dx}{a^3}-\frac{\left (b \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^3}\\ &=\frac{d \cos (c+d x)}{2 a b (a+b x)}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^3}+\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{\sin (c+d x)}{a^2 (a+b x)}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{d^2 \int \frac{\sin (c+d x)}{a+b x} \, dx}{2 a b}-\frac{\left (d \cos \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{\left (d \sin \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{d \cos (c+d x)}{2 a b (a+b x)}-\frac{d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{a^2 b}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^3}+\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{\sin (c+d x)}{a^2 (a+b x)}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^2 b}+\frac{\left (d^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}{2 a b}+\frac{\left (d^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}{2 a b}\\ &=\frac{d \cos (c+d x)}{2 a b (a+b x)}-\frac{d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{a^2 b}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^3}+\frac{d^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{2 a b^2}+\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{\sin (c+d x)}{a^2 (a+b x)}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{2 a b^2}+\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^2 b}\\ \end{align*}

Mathematica [C]  time = 11.7937, size = 1749, normalized size = 6.7 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]/(x*(a + b*x)^3),x]

[Out]

((3*I)*a^2*b^2*E^(((2*I)*a*d)/b)*Cos[c] + a^3*b*d*E^(((2*I)*a*d)/b)*Cos[c] - (3*I)*a^2*b^2*E^(((2*I)*d*(a + b*
x))/b)*Cos[c] + a^3*b*d*E^(((2*I)*d*(a + b*x))/b)*Cos[c] + (2*I)*a*b^3*E^(((2*I)*a*d)/b)*x*Cos[c] + a^2*b^2*d*
E^(((2*I)*a*d)/b)*x*Cos[c] - (2*I)*a*b^3*E^(((2*I)*d*(a + b*x))/b)*x*Cos[c] + a^2*b^2*d*E^(((2*I)*d*(a + b*x))
/b)*x*Cos[c] - 2*a^3*b*d*E^((I*d*(3*a + b*x))/b)*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] + I*a^4*d^2*E^((I*
d*(3*a + b*x))/b)*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] - 4*a^2*b^2*d*E^((I*d*(3*a + b*x))/b)*x*Cos[c]*Ex
pIntegralEi[((-I)*d*(a + b*x))/b] + (2*I)*a^3*b*d^2*E^((I*d*(3*a + b*x))/b)*x*Cos[c]*ExpIntegralEi[((-I)*d*(a
+ b*x))/b] - 2*a*b^3*d*E^((I*d*(3*a + b*x))/b)*x^2*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] + I*a^2*b^2*d^2*
E^((I*d*(3*a + b*x))/b)*x^2*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] - 2*a^3*b*d*E^((I*d*(a + b*x))/b)*Cos[c
]*ExpIntegralEi[(I*d*(a + b*x))/b] - I*a^4*d^2*E^((I*d*(a + b*x))/b)*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] -
4*a^2*b^2*d*E^((I*d*(a + b*x))/b)*x*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] - (2*I)*a^3*b*d^2*E^((I*d*(a + b*
x))/b)*x*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] - 2*a*b^3*d*E^((I*d*(a + b*x))/b)*x^2*Cos[c]*ExpIntegralEi[(I
*d*(a + b*x))/b] - I*a^2*b^2*d^2*E^((I*d*(a + b*x))/b)*x^2*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] + 3*a^2*b^2
*E^(((2*I)*a*d)/b)*Sin[c] - I*a^3*b*d*E^(((2*I)*a*d)/b)*Sin[c] + 3*a^2*b^2*E^(((2*I)*d*(a + b*x))/b)*Sin[c] +
I*a^3*b*d*E^(((2*I)*d*(a + b*x))/b)*Sin[c] + 2*a*b^3*E^(((2*I)*a*d)/b)*x*Sin[c] - I*a^2*b^2*d*E^(((2*I)*a*d)/b
)*x*Sin[c] + 2*a*b^3*E^(((2*I)*d*(a + b*x))/b)*x*Sin[c] + I*a^2*b^2*d*E^(((2*I)*d*(a + b*x))/b)*x*Sin[c] + 4*b
^2*E^((I*d*(2*a + b*x))/b)*(a + b*x)^2*CosIntegral[d*x]*Sin[c] + (2*I)*a^3*b*d*E^((I*d*(3*a + b*x))/b)*ExpInte
gralEi[((-I)*d*(a + b*x))/b]*Sin[c] + a^4*d^2*E^((I*d*(3*a + b*x))/b)*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[
c] + (4*I)*a^2*b^2*d*E^((I*d*(3*a + b*x))/b)*x*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] + 2*a^3*b*d^2*E^((I*
d*(3*a + b*x))/b)*x*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] + (2*I)*a*b^3*d*E^((I*d*(3*a + b*x))/b)*x^2*Exp
IntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] + a^2*b^2*d^2*E^((I*d*(3*a + b*x))/b)*x^2*ExpIntegralEi[((-I)*d*(a + b
*x))/b]*Sin[c] - (2*I)*a^3*b*d*E^((I*d*(a + b*x))/b)*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] + a^4*d^2*E^((I*d
*(a + b*x))/b)*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] - (4*I)*a^2*b^2*d*E^((I*d*(a + b*x))/b)*x*ExpIntegralEi
[(I*d*(a + b*x))/b]*Sin[c] + 2*a^3*b*d^2*E^((I*d*(a + b*x))/b)*x*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] - (2*
I)*a*b^3*d*E^((I*d*(a + b*x))/b)*x^2*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] + a^2*b^2*d^2*E^((I*d*(a + b*x))/
b)*x^2*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] - 4*b^2*E^((I*d*(2*a + b*x))/b)*(a + b*x)^2*CosIntegral[d*(a/b
+ x)]*Sin[c - (a*d)/b] + 4*a^2*b^2*E^((I*d*(2*a + b*x))/b)*Cos[c]*SinIntegral[d*x] + 8*a*b^3*E^((I*d*(2*a + b*
x))/b)*x*Cos[c]*SinIntegral[d*x] + 4*b^4*E^((I*d*(2*a + b*x))/b)*x^2*Cos[c]*SinIntegral[d*x] - 4*a^2*b^2*E^((I
*d*(2*a + b*x))/b)*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 8*a*b^3*E^((I*d*(2*a + b*x))/b)*x*Cos[c - (a*d)
/b]*SinIntegral[d*(a/b + x)] - 4*b^4*E^((I*d*(2*a + b*x))/b)*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(4
*a^3*b^2*E^((I*d*(2*a + b*x))/b)*(a + b*x)^2)

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Maple [A]  time = 0.012, size = 359, normalized size = 1.4 \begin{align*} -{\frac{{d}^{2}b}{a} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\, \left ( \left ( dx+c \right ) b+da-cb \right ) ^{2}b}}+{\frac{1}{2\,b} \left ( -{\frac{\cos \left ( dx+c \right ) }{ \left ( \left ( dx+c \right ) b+da-cb \right ) b}}-{\frac{1}{b} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) } \right ) }-{\frac{bd}{{a}^{2}} \left ( -{\frac{\sin \left ( dx+c \right ) }{ \left ( \left ( dx+c \right ) b+da-cb \right ) b}}+{\frac{1}{b} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) }+{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) }-{\frac{b}{{a}^{3}} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) }+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x/(b*x+a)^3,x)

[Out]

-d^2*b/a*(-1/2*sin(d*x+c)/((d*x+c)*b+d*a-c*b)^2/b+1/2*(-cos(d*x+c)/((d*x+c)*b+d*a-c*b)/b-(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)/b)/b)-d*b/a^2*(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/
b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-b/a^3*(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)+1/a^3*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x + a\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x/(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x + a)^3*x), x)

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Fricas [B]  time = 1.66022, size = 1212, normalized size = 4.64 \begin{align*} \frac{4 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \cos \left (c\right ) \operatorname{Si}\left (d x\right ) + 2 \,{\left (a^{2} b^{2} d x + a^{3} b d\right )} \cos \left (d x + c\right ) - 2 \,{\left ({\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) -{\left (a^{4} d^{2} - 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) + 2 \,{\left (2 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \sin \left (d x + c\right ) + 2 \,{\left ({\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \operatorname{Ci}\left (d x\right ) +{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right ) -{\left ({\left (a^{4} d^{2} - 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{4} d^{2} - 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) + 4 \,{\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{4 \,{\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(4*(b^4*x^2 + 2*a*b^3*x + a^2*b^2)*cos(c)*sin_integral(d*x) + 2*(a^2*b^2*d*x + a^3*b*d)*cos(d*x + c) - 2*(
(a*b^3*d*x^2 + 2*a^2*b^2*d*x + a^3*b*d)*cos_integral((b*d*x + a*d)/b) + (a*b^3*d*x^2 + 2*a^2*b^2*d*x + a^3*b*d
)*cos_integral(-(b*d*x + a*d)/b) - (a^4*d^2 - 2*a^2*b^2 + (a^2*b^2*d^2 - 2*b^4)*x^2 + 2*(a^3*b*d^2 - 2*a*b^3)*
x)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) + 2*(2*a*b^3*x + 3*a^2*b^2)*sin(d*x + c) + 2*((b^4*x^2 +
2*a*b^3*x + a^2*b^2)*cos_integral(d*x) + (b^4*x^2 + 2*a*b^3*x + a^2*b^2)*cos_integral(-d*x))*sin(c) - ((a^4*d
^2 - 2*a^2*b^2 + (a^2*b^2*d^2 - 2*b^4)*x^2 + 2*(a^3*b*d^2 - 2*a*b^3)*x)*cos_integral((b*d*x + a*d)/b) + (a^4*d
^2 - 2*a^2*b^2 + (a^2*b^2*d^2 - 2*b^4)*x^2 + 2*(a^3*b*d^2 - 2*a*b^3)*x)*cos_integral(-(b*d*x + a*d)/b) + 4*(a*
b^3*d*x^2 + 2*a^2*b^2*d*x + a^3*b*d)*sin_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(a^3*b^4*x^2 + 2*a^4*
b^3*x + a^5*b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (c + d x \right )}}{x \left (a + b x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x/(b*x+a)**3,x)

[Out]

Integral(sin(c + d*x)/(x*(a + b*x)**3), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x/(b*x+a)^3,x, algorithm="giac")

[Out]

Timed out