3.38 \(\int \frac{\sin (c+d x)}{x^2 (a+b x)^3} \, dx\)

Optimal. Leaf size=299 \[ -\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{2 a^2 b}-\frac{3 b \sin (c) \text{CosIntegral}(d x)}{a^4}+\frac{3 b \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{2 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}-\frac{2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}+\frac{3 b \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^4}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{d \cos (c+d x)}{2 a^2 (a+b x)}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a^3}-\frac{d \sin (c) \text{Si}(d x)}{a^3}-\frac{\sin (c+d x)}{a^3 x} \]

[Out]

-(d*Cos[c + d*x])/(2*a^2*(a + b*x)) + (d*Cos[c]*CosIntegral[d*x])/a^3 + (2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d
)/b + d*x])/a^3 - (3*b*CosIntegral[d*x]*Sin[c])/a^4 + (3*b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^4 -
(d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*a^2*b) - Sin[c + d*x]/(a^3*x) - (b*Sin[c + d*x])/(2*a^2*(
a + b*x)^2) - (2*b*Sin[c + d*x])/(a^3*(a + b*x)) - (3*b*Cos[c]*SinIntegral[d*x])/a^4 - (d*Sin[c]*SinIntegral[d
*x])/a^3 + (3*b*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^4 - (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b +
d*x])/(2*a^2*b) - (2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3

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Rubi [A]  time = 0.667508, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3299, 3302} \[ -\frac{d^2 \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{2 a^2 b}-\frac{3 b \sin (c) \text{CosIntegral}(d x)}{a^4}+\frac{3 b \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{2 d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}-\frac{2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}+\frac{3 b \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^4}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{d \cos (c+d x)}{2 a^2 (a+b x)}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a^3}-\frac{d \sin (c) \text{Si}(d x)}{a^3}-\frac{\sin (c+d x)}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^2*(a + b*x)^3),x]

[Out]

-(d*Cos[c + d*x])/(2*a^2*(a + b*x)) + (d*Cos[c]*CosIntegral[d*x])/a^3 + (2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d
)/b + d*x])/a^3 - (3*b*CosIntegral[d*x]*Sin[c])/a^4 + (3*b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^4 -
(d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*a^2*b) - Sin[c + d*x]/(a^3*x) - (b*Sin[c + d*x])/(2*a^2*(
a + b*x)^2) - (2*b*Sin[c + d*x])/(a^3*(a + b*x)) - (3*b*Cos[c]*SinIntegral[d*x])/a^4 - (d*Sin[c]*SinIntegral[d
*x])/a^3 + (3*b*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^4 - (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b +
d*x])/(2*a^2*b) - (2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3

Rule 6742

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

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]

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]

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{x^2 (a+b x)^3} \, dx &=\int \left (\frac{\sin (c+d x)}{a^3 x^2}-\frac{3 b \sin (c+d x)}{a^4 x}+\frac{b^2 \sin (c+d x)}{a^2 (a+b x)^3}+\frac{2 b^2 \sin (c+d x)}{a^3 (a+b x)^2}+\frac{3 b^2 \sin (c+d x)}{a^4 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x^2} \, dx}{a^3}-\frac{(3 b) \int \frac{\sin (c+d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2\right ) \int \frac{\sin (c+d x)}{a+b x} \, dx}{a^4}+\frac{\left (2 b^2\right ) \int \frac{\sin (c+d x)}{(a+b x)^2} \, dx}{a^3}+\frac{b^2 \int \frac{\sin (c+d x)}{(a+b x)^3} \, dx}{a^2}\\ &=-\frac{\sin (c+d x)}{a^3 x}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}+\frac{d \int \frac{\cos (c+d x)}{x} \, dx}{a^3}+\frac{(2 b d) \int \frac{\cos (c+d x)}{a+b x} \, dx}{a^3}+\frac{(b d) \int \frac{\cos (c+d x)}{(a+b x)^2} \, dx}{2 a^2}-\frac{(3 b \cos (c)) \int \frac{\sin (d x)}{x} \, dx}{a^4}+\frac{\left (3 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^4}-\frac{(3 b \sin (c)) \int \frac{\cos (d x)}{x} \, dx}{a^4}+\frac{\left (3 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^4}\\ &=-\frac{d \cos (c+d x)}{2 a^2 (a+b x)}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}+\frac{3 b \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^4}-\frac{\sin (c+d x)}{a^3 x}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}+\frac{3 b \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^4}-\frac{d^2 \int \frac{\sin (c+d x)}{a+b x} \, dx}{2 a^2}+\frac{(d \cos (c)) \int \frac{\cos (d x)}{x} \, dx}{a^3}+\frac{\left (2 b 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^3}-\frac{(d \sin (c)) \int \frac{\sin (d x)}{x} \, dx}{a^3}-\frac{\left (2 b 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^3}\\ &=-\frac{d \cos (c+d x)}{2 a^2 (a+b x)}+\frac{d \cos (c) \text{Ci}(d x)}{a^3}+\frac{2 d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}+\frac{3 b \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^4}-\frac{\sin (c+d x)}{a^3 x}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d \sin (c) \text{Si}(d x)}{a^3}+\frac{3 b \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^4}-\frac{2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}-\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^2}-\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^2}\\ &=-\frac{d \cos (c+d x)}{2 a^2 (a+b x)}+\frac{d \cos (c) \text{Ci}(d x)}{a^3}+\frac{2 d \cos \left (c-\frac{a d}{b}\right ) \text{Ci}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}+\frac{3 b \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{a^4}-\frac{d^2 \text{Ci}\left (\frac{a d}{b}+d x\right ) \sin \left (c-\frac{a d}{b}\right )}{2 a^2 b}-\frac{\sin (c+d x)}{a^3 x}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d \sin (c) \text{Si}(d x)}{a^3}+\frac{3 b \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^4}-\frac{d^2 \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{2 a^2 b}-\frac{2 d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (\frac{a d}{b}+d x\right )}{a^3}\\ \end{align*}

Mathematica [C]  time = 6.00374, size = 2108, normalized size = 7.05 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]/(x^2*(a + b*x)^3),x]

[Out]

((-4*I)*a^2*b^2*E^(((2*I)*a*d)/b)*x*Cos[c] - a^3*b*d*E^(((2*I)*a*d)/b)*x*Cos[c] + (4*I)*a^2*b^2*E^(((2*I)*d*(a
+ b*x))/b)*x*Cos[c] - a^3*b*d*E^(((2*I)*d*(a + b*x))/b)*x*Cos[c] - (4*I)*a*b^3*E^(((2*I)*a*d)/b)*x^2*Cos[c] -
a^2*b^2*d*E^(((2*I)*a*d)/b)*x^2*Cos[c] + (4*I)*a*b^3*E^(((2*I)*d*(a + b*x))/b)*x^2*Cos[c] - a^2*b^2*d*E^(((2*
I)*d*(a + b*x))/b)*x^2*Cos[c] + 4*a^3*b*d*E^((I*d*(3*a + b*x))/b)*x*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b]
- I*a^4*d^2*E^((I*d*(3*a + b*x))/b)*x*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] + 8*a^2*b^2*d*E^((I*d*(3*a +
b*x))/b)*x^2*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] - (2*I)*a^3*b*d^2*E^((I*d*(3*a + b*x))/b)*x^2*Cos[c]*
ExpIntegralEi[((-I)*d*(a + b*x))/b] + 4*a*b^3*d*E^((I*d*(3*a + b*x))/b)*x^3*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^3*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x))/b] + 4*a^3*b*d*E^
((I*d*(a + b*x))/b)*x*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] + I*a^4*d^2*E^((I*d*(a + b*x))/b)*x*Cos[c]*ExpIn
tegralEi[(I*d*(a + b*x))/b] + 8*a^2*b^2*d*E^((I*d*(a + b*x))/b)*x^2*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] +
(2*I)*a^3*b*d^2*E^((I*d*(a + b*x))/b)*x^2*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] + 4*a*b^3*d*E^((I*d*(a + b*x
))/b)*x^3*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] + I*a^2*b^2*d^2*E^((I*d*(a + b*x))/b)*x^3*Cos[c]*ExpIntegral
Ei[(I*d*(a + b*x))/b] - 4*a^2*b^2*E^(((2*I)*a*d)/b)*x*Sin[c] + I*a^3*b*d*E^(((2*I)*a*d)/b)*x*Sin[c] - 4*a^2*b^
2*E^(((2*I)*d*(a + b*x))/b)*x*Sin[c] - I*a^3*b*d*E^(((2*I)*d*(a + b*x))/b)*x*Sin[c] - 4*a*b^3*E^(((2*I)*a*d)/b
)*x^2*Sin[c] + I*a^2*b^2*d*E^(((2*I)*a*d)/b)*x^2*Sin[c] - 4*a*b^3*E^(((2*I)*d*(a + b*x))/b)*x^2*Sin[c] - I*a^2
*b^2*d*E^(((2*I)*d*(a + b*x))/b)*x^2*Sin[c] - 4*a^3*b*E^((I*d*(2*a + b*x))/b)*Cos[d*x]*Sin[c] - 10*a^2*b^2*E^(
(I*d*(2*a + b*x))/b)*x*Cos[d*x]*Sin[c] - 4*a*b^3*E^((I*d*(2*a + b*x))/b)*x^2*Cos[d*x]*Sin[c] - (4*I)*a^3*b*d*E
^((I*d*(3*a + b*x))/b)*x*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] - a^4*d^2*E^((I*d*(3*a + b*x))/b)*x*ExpInt
egralEi[((-I)*d*(a + b*x))/b]*Sin[c] - (8*I)*a^2*b^2*d*E^((I*d*(3*a + b*x))/b)*x^2*ExpIntegralEi[((-I)*d*(a +
b*x))/b]*Sin[c] - 2*a^3*b*d^2*E^((I*d*(3*a + b*x))/b)*x^2*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] - (4*I)*a
*b^3*d*E^((I*d*(3*a + b*x))/b)*x^3*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] - a^2*b^2*d^2*E^((I*d*(3*a + b*x
))/b)*x^3*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] + (4*I)*a^3*b*d*E^((I*d*(a + b*x))/b)*x*ExpIntegralEi[(I*
d*(a + b*x))/b]*Sin[c] - a^4*d^2*E^((I*d*(a + b*x))/b)*x*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] + (8*I)*a^2*b
^2*d*E^((I*d*(a + b*x))/b)*x^2*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] - 2*a^3*b*d^2*E^((I*d*(a + b*x))/b)*x^2
*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] + (4*I)*a*b^3*d*E^((I*d*(a + b*x))/b)*x^3*ExpIntegralEi[(I*d*(a + b*x
))/b]*Sin[c] - a^2*b^2*d^2*E^((I*d*(a + b*x))/b)*x^3*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] + 4*b*E^((I*d*(2*
a + b*x))/b)*x*(a + b*x)^2*CosIntegral[d*x]*(a*d*Cos[c] - 3*b*Sin[c]) + 12*b^2*E^((I*d*(2*a + b*x))/b)*x*(a +
b*x)^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] - 4*a^3*b*E^((I*d*(2*a + b*x))/b)*Cos[c]*Sin[d*x] - 10*a^2*b^
2*E^((I*d*(2*a + b*x))/b)*x*Cos[c]*Sin[d*x] - 4*a*b^3*E^((I*d*(2*a + b*x))/b)*x^2*Cos[c]*Sin[d*x] - 12*a^2*b^2
*E^((I*d*(2*a + b*x))/b)*x*Cos[c]*SinIntegral[d*x] - 24*a*b^3*E^((I*d*(2*a + b*x))/b)*x^2*Cos[c]*SinIntegral[d
*x] - 12*b^4*E^((I*d*(2*a + b*x))/b)*x^3*Cos[c]*SinIntegral[d*x] - 4*a^3*b*d*E^((I*d*(2*a + b*x))/b)*x*Sin[c]*
SinIntegral[d*x] - 8*a^2*b^2*d*E^((I*d*(2*a + b*x))/b)*x^2*Sin[c]*SinIntegral[d*x] - 4*a*b^3*d*E^((I*d*(2*a +
b*x))/b)*x^3*Sin[c]*SinIntegral[d*x] + 12*a^2*b^2*E^((I*d*(2*a + b*x))/b)*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/
b + x)] + 24*a*b^3*E^((I*d*(2*a + b*x))/b)*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 12*b^4*E^((I*d*(2*a
+ b*x))/b)*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(4*a^4*b*E^((I*d*(2*a + b*x))/b)*x*(a + b*x)^2)

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Maple [A]  time = 0.012, size = 405, normalized size = 1.4 \begin{align*} d \left ({\frac{{b}^{2}d}{{a}^{2}} \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{1}{{a}^{3}} \left ( -{\frac{\sin \left ( dx+c \right ) }{dx}}-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) \right ) }+2\,{\frac{{b}^{2}}{{a}^{3}} \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 ) }+3\,{\frac{{b}^{2}}{d{a}^{4}} \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 ) }-3\,{\frac{b \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{d{a}^{4}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x^2/(b*x+a)^3,x)

[Out]

d*(d*b^2/a^2*(-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)+1/a^3*(-sin(d*x+c)/x/d-Si(d*x)*sin(c)
+Ci(d*x)*cos(c))+2*b^2/a^3*(-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)+3/d*b^2/a^4*(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)-3/d/a^4*b*(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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^2/(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x + a)^3*x^2), x)

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Fricas [B]  time = 1.70193, size = 1577, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/4*(2*(a^2*b^2*d*x^2 + a^3*b*d*x)*cos(d*x + c) - 2*((a*b^3*d*x^3 + 2*a^2*b^2*d*x^2 + a^3*b*d*x)*cos_integral
(d*x) + (a*b^3*d*x^3 + 2*a^2*b^2*d*x^2 + a^3*b*d*x)*cos_integral(-d*x) - 6*(b^4*x^3 + 2*a*b^3*x^2 + a^2*b^2*x)
*sin_integral(d*x))*cos(c) - 2*(2*(a*b^3*d*x^3 + 2*a^2*b^2*d*x^2 + a^3*b*d*x)*cos_integral((b*d*x + a*d)/b) +
2*(a*b^3*d*x^3 + 2*a^2*b^2*d*x^2 + a^3*b*d*x)*cos_integral(-(b*d*x + a*d)/b) - ((a^2*b^2*d^2 - 6*b^4)*x^3 + 2*
(a^3*b*d^2 - 6*a*b^3)*x^2 + (a^4*d^2 - 6*a^2*b^2)*x)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) + 2*(6
*a*b^3*x^2 + 9*a^2*b^2*x + 2*a^3*b)*sin(d*x + c) + 2*(3*(b^4*x^3 + 2*a*b^3*x^2 + a^2*b^2*x)*cos_integral(d*x)
+ 3*(b^4*x^3 + 2*a*b^3*x^2 + a^2*b^2*x)*cos_integral(-d*x) + 2*(a*b^3*d*x^3 + 2*a^2*b^2*d*x^2 + a^3*b*d*x)*sin
_integral(d*x))*sin(c) - (((a^2*b^2*d^2 - 6*b^4)*x^3 + 2*(a^3*b*d^2 - 6*a*b^3)*x^2 + (a^4*d^2 - 6*a^2*b^2)*x)*
cos_integral((b*d*x + a*d)/b) + ((a^2*b^2*d^2 - 6*b^4)*x^3 + 2*(a^3*b*d^2 - 6*a*b^3)*x^2 + (a^4*d^2 - 6*a^2*b^
2)*x)*cos_integral(-(b*d*x + a*d)/b) + 8*(a*b^3*d*x^3 + 2*a^2*b^2*d*x^2 + a^3*b*d*x)*sin_integral((b*d*x + a*d
)/b))*sin(-(b*c - a*d)/b))/(a^4*b^3*x^3 + 2*a^5*b^2*x^2 + a^6*b*x)

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Sympy [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**2/(b*x+a)**3,x)

[Out]

Timed out

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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^2/(b*x+a)^3,x, algorithm="giac")

[Out]

Timed out