\(\int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx\) [39]

Optimal result

Integrand size = 17, antiderivative size = 377 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=-\frac {d \cos (c+d x)}{2 a^3 x}+\frac {b d \cos (c+d x)}{2 a^3 (a+b x)}-\frac {3 b d \cos (c) \operatorname {CosIntegral}(d x)}{a^4}-\frac {3 b d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^4}+\frac {6 b^2 \operatorname {CosIntegral}(d x) \sin (c)}{a^5}-\frac {d^2 \operatorname {CosIntegral}(d x) \sin (c)}{2 a^3}-\frac {6 b^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^5}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 a^3}-\frac {\sin (c+d x)}{2 a^3 x^2}+\frac {3 b \sin (c+d x)}{a^4 x}+\frac {b^2 \sin (c+d x)}{2 a^3 (a+b x)^2}+\frac {3 b^2 \sin (c+d x)}{a^4 (a+b x)}+\frac {6 b^2 \cos (c) \text {Si}(d x)}{a^5}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}+\frac {3 b d \sin (c) \text {Si}(d x)}{a^4}-\frac {6 b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^5}+\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 a^3}+\frac {3 b d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^4} \] Output:

-1/2*d*cos(d*x+c)/a^3/x+1/2*b*d*cos(d*x+c)/a^3/(b*x+a)-3*b*d*cos(c)*Ci(d*x 
)/a^4-3*b*d*cos(-c+a*d/b)*Ci(a*d/b+d*x)/a^4+6*b^2*Ci(d*x)*sin(c)/a^5-1/2*d 
^2*Ci(d*x)*sin(c)/a^3+6*b^2*Ci(a*d/b+d*x)*sin(-c+a*d/b)/a^5-1/2*d^2*Ci(a*d 
/b+d*x)*sin(-c+a*d/b)/a^3-1/2*sin(d*x+c)/a^3/x^2+3*b*sin(d*x+c)/a^4/x+1/2* 
b^2*sin(d*x+c)/a^3/(b*x+a)^2+3*b^2*sin(d*x+c)/a^4/(b*x+a)+6*b^2*cos(c)*Si( 
d*x)/a^5-1/2*d^2*cos(c)*Si(d*x)/a^3+3*b*d*sin(c)*Si(d*x)/a^4-6*b^2*cos(-c+ 
a*d/b)*Si(a*d/b+d*x)/a^5+1/2*d^2*cos(-c+a*d/b)*Si(a*d/b+d*x)/a^3-3*b*d*sin 
(-c+a*d/b)*Si(a*d/b+d*x)/a^4
 

Mathematica [A] (verified)

Time = 2.11 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.67 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=\frac {-a^4 d x \cos (c+d x)-a^3 b d x^2 \cos (c+d x)-x^2 (a+b x)^2 \operatorname {CosIntegral}(d x) \left (6 a b d \cos (c)+\left (-12 b^2+a^2 d^2\right ) \sin (c)\right )+x^2 (a+b x)^2 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-6 a b d \cos \left (c-\frac {a d}{b}\right )+\left (-12 b^2+a^2 d^2\right ) \sin \left (c-\frac {a d}{b}\right )\right )-a^4 \sin (c+d x)+4 a^3 b x \sin (c+d x)+18 a^2 b^2 x^2 \sin (c+d x)+12 a b^3 x^3 \sin (c+d x)+12 a^2 b^2 x^2 \cos (c) \text {Si}(d x)-a^4 d^2 x^2 \cos (c) \text {Si}(d x)+24 a b^3 x^3 \cos (c) \text {Si}(d x)-2 a^3 b d^2 x^3 \cos (c) \text {Si}(d x)+12 b^4 x^4 \cos (c) \text {Si}(d x)-a^2 b^2 d^2 x^4 \cos (c) \text {Si}(d x)+6 a^3 b d x^2 \sin (c) \text {Si}(d x)+12 a^2 b^2 d x^3 \sin (c) \text {Si}(d x)+6 a b^3 d x^4 \sin (c) \text {Si}(d x)-12 a^2 b^2 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a^4 d^2 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )-24 a b^3 x^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d^2 x^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )-12 b^4 x^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a^2 b^2 d^2 x^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+6 a^3 b d x^2 \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+12 a^2 b^2 d x^3 \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+6 a b^3 d x^4 \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^5 x^2 (a+b x)^2} \] Input:

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

Output:

(-(a^4*d*x*Cos[c + d*x]) - a^3*b*d*x^2*Cos[c + d*x] - x^2*(a + b*x)^2*CosI 
ntegral[d*x]*(6*a*b*d*Cos[c] + (-12*b^2 + a^2*d^2)*Sin[c]) + x^2*(a + b*x) 
^2*CosIntegral[d*(a/b + x)]*(-6*a*b*d*Cos[c - (a*d)/b] + (-12*b^2 + a^2*d^ 
2)*Sin[c - (a*d)/b]) - a^4*Sin[c + d*x] + 4*a^3*b*x*Sin[c + d*x] + 18*a^2* 
b^2*x^2*Sin[c + d*x] + 12*a*b^3*x^3*Sin[c + d*x] + 12*a^2*b^2*x^2*Cos[c]*S 
inIntegral[d*x] - a^4*d^2*x^2*Cos[c]*SinIntegral[d*x] + 24*a*b^3*x^3*Cos[c 
]*SinIntegral[d*x] - 2*a^3*b*d^2*x^3*Cos[c]*SinIntegral[d*x] + 12*b^4*x^4* 
Cos[c]*SinIntegral[d*x] - a^2*b^2*d^2*x^4*Cos[c]*SinIntegral[d*x] + 6*a^3* 
b*d*x^2*Sin[c]*SinIntegral[d*x] + 12*a^2*b^2*d*x^3*Sin[c]*SinIntegral[d*x] 
 + 6*a*b^3*d*x^4*Sin[c]*SinIntegral[d*x] - 12*a^2*b^2*x^2*Cos[c - (a*d)/b] 
*SinIntegral[d*(a/b + x)] + a^4*d^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/ 
b + x)] - 24*a*b^3*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 2*a^3*b 
*d^2*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 12*b^4*x^4*Cos[c - (a 
*d)/b]*SinIntegral[d*(a/b + x)] + a^2*b^2*d^2*x^4*Cos[c - (a*d)/b]*SinInte 
gral[d*(a/b + x)] + 6*a^3*b*d*x^2*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x) 
] + 12*a^2*b^2*d*x^3*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 6*a*b^3*d 
*x^4*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(2*a^5*x^2*(a + b*x)^2)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {6 b^3 \sin (c+d x)}{a^5 (a+b x)}+\frac {6 b^2 \sin (c+d x)}{a^5 x}-\frac {3 b^3 \sin (c+d x)}{a^4 (a+b x)^2}-\frac {3 b \sin (c+d x)}{a^4 x^2}-\frac {b^3 \sin (c+d x)}{a^3 (a+b x)^3}+\frac {\sin (c+d x)}{a^3 x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 b^2 \sin (c) \operatorname {CosIntegral}(d x)}{a^5}-\frac {6 b^2 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a^5}+\frac {6 b^2 \cos (c) \text {Si}(d x)}{a^5}-\frac {6 b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^5}+\frac {3 b^2 \sin (c+d x)}{a^4 (a+b x)}-\frac {3 b d \cos (c) \operatorname {CosIntegral}(d x)}{a^4}-\frac {3 b d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{a^4}+\frac {3 b d \sin (c) \text {Si}(d x)}{a^4}+\frac {3 b d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^4}+\frac {3 b \sin (c+d x)}{a^4 x}+\frac {b^2 \sin (c+d x)}{2 a^3 (a+b x)^2}+\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{2 a^3}+\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 a^3}+\frac {b d \cos (c+d x)}{2 a^3 (a+b x)}-\frac {d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a^3}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {d \cos (c+d x)}{2 a^3 x}\)

Input:

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

Output:

-1/2*(d*Cos[c + d*x])/(a^3*x) + (b*d*Cos[c + d*x])/(2*a^3*(a + b*x)) - (3* 
b*d*Cos[c]*CosIntegral[d*x])/a^4 - (3*b*d*Cos[c - (a*d)/b]*CosIntegral[(a* 
d)/b + d*x])/a^4 + (6*b^2*CosIntegral[d*x]*Sin[c])/a^5 - (d^2*CosIntegral[ 
d*x]*Sin[c])/(2*a^3) - (6*b^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b]) 
/a^5 + (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*a^3) - Sin[c + 
 d*x]/(2*a^3*x^2) + (3*b*Sin[c + d*x])/(a^4*x) + (b^2*Sin[c + d*x])/(2*a^3 
*(a + b*x)^2) + (3*b^2*Sin[c + d*x])/(a^4*(a + b*x)) + (6*b^2*Cos[c]*SinIn 
tegral[d*x])/a^5 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a^3) + (3*b*d*Sin[c]*S 
inIntegral[d*x])/a^4 - (6*b^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x]) 
/a^5 + (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*a^3) + (3*b*d* 
Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

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

Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(d^{2} \left (\frac {6 b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{2} a^{5}}-\frac {b^{3} \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )}{a^{3}}-\frac {6 b^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{d^{2} a^{5}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}}{a^{3}}-\frac {3 b^{3} \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{d \,a^{4}}-\frac {3 b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d \,a^{4}}\right )\)	\(466\)
default	\(d^{2} \left (\frac {6 b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{2} a^{5}}-\frac {b^{3} \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )}{a^{3}}-\frac {6 b^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{d^{2} a^{5}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}}{a^{3}}-\frac {3 b^{3} \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{d \,a^{4}}-\frac {3 b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d \,a^{4}}\right )\)	\(466\)
risch	\(\frac {3 d b \,{\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a^{4}}+\frac {3 i b^{2} {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{a^{5}}-\frac {3 i b^{2} {\mathrm e}^{-\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {i a d -i b c}{b}\right )}{a^{5}}+\frac {i d^{2} {\mathrm e}^{-\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {i a d -i b c}{b}\right )}{4 a^{3}}+\frac {3 d b \,{\mathrm e}^{-\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{4}}-\frac {i d^{2} {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{4 a^{3}}+\frac {3 d \,{\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (i d x \right ) b}{2 a^{4}}+\frac {i d^{2} {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (i d x \right )}{4 a^{3}}-\frac {3 i {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (i d x \right ) b^{2}}{a^{5}}+\frac {3 d \,{\mathrm e}^{\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (i d x +i c +\frac {i \left (a d -b c \right )}{b}\right ) b}{2 a^{4}}-\frac {i d^{2} {\mathrm e}^{\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (i d x +i c +\frac {i \left (a d -b c \right )}{b}\right )}{4 a^{3}}+\frac {3 i {\mathrm e}^{\frac {i \left (a d -b c \right )}{b}} \operatorname {expIntegral}_{1}\left (i d x +i c +\frac {i \left (a d -b c \right )}{b}\right ) b^{2}}{a^{5}}-\frac {i \left (2 i a^{2} b^{3} d^{5} x^{6}+6 i a^{3} b^{2} d^{5} x^{5}+6 i a^{4} b \,d^{5} x^{4}+2 i a^{5} d^{5} x^{3}\right ) \cos \left (d x +c \right )}{4 a^{4} x^{4} \left (b x +a \right )^{2} d^{2} \left (-x^{2} d^{2} b^{2}-2 a b \,d^{2} x -a^{2} d^{2}\right )}+\frac {\left (-24 b^{5} d^{4} x^{7}-84 a \,b^{4} d^{4} x^{6}-104 a^{2} b^{3} d^{4} x^{5}-50 a^{3} b^{2} d^{4} x^{4}-4 a^{4} b \,d^{4} x^{3}+2 a^{5} d^{4} x^{2}\right ) \sin \left (d x +c \right )}{4 a^{4} x^{4} \left (b x +a \right )^{2} d^{2} \left (-x^{2} d^{2} b^{2}-2 a b \,d^{2} x -a^{2} d^{2}\right )}\)	\(666\)

Input:

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

Output:

d^2*(6/d^2/a^5*b^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))-b^3/a^3*(-1/2*sin(d*x+c 
)/(a*d-b*c+b*(d*x+c))^2/b+1/2*(-cos(d*x+c)/(a*d-b*c+b*(d*x+c))/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)-6/d^2*b^3/a^5*(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*(-1/2*sin(d*x+c)/d^2/x^2-1/2*cos(d*x+ 
c)/d/x-1/2*Si(d*x)*cos(c)-1/2*Ci(d*x)*sin(c))-3/d*b^3/a^4*(-sin(d*x+c)/(a* 
d-b*c+b*(d*x+c))/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/a^4*b*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+ 
Ci(d*x)*cos(c)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.55 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=-\frac {{\left (a^{3} b d x^{2} + a^{4} d x\right )} \cos \left (d x + c\right ) + {\left (6 \, {\left (a b^{3} d x^{4} + 2 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2}\right )} \operatorname {Ci}\left (d x\right ) + {\left ({\left (a^{2} b^{2} d^{2} - 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 12 \, a^{2} b^{2}\right )} x^{2}\right )} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) + {\left (6 \, {\left (a b^{3} d x^{4} + 2 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - {\left ({\left (a^{2} b^{2} d^{2} - 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 12 \, a^{2} b^{2}\right )} x^{2}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left (12 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - a^{4}\right )} \sin \left (d x + c\right ) + {\left ({\left ({\left (a^{2} b^{2} d^{2} - 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 12 \, a^{2} b^{2}\right )} x^{2}\right )} \operatorname {Ci}\left (d x\right ) - 6 \, {\left (a b^{3} d x^{4} + 2 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2}\right )} \operatorname {Si}\left (d x\right )\right )} \sin \left (c\right ) + {\left ({\left ({\left (a^{2} b^{2} d^{2} - 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 12 \, a^{2} b^{2}\right )} x^{2}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 6 \, {\left (a b^{3} d x^{4} + 2 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \] Input:

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

Output:

-1/2*((a^3*b*d*x^2 + a^4*d*x)*cos(d*x + c) + (6*(a*b^3*d*x^4 + 2*a^2*b^2*d 
*x^3 + a^3*b*d*x^2)*cos_integral(d*x) + ((a^2*b^2*d^2 - 12*b^4)*x^4 + 2*(a 
^3*b*d^2 - 12*a*b^3)*x^3 + (a^4*d^2 - 12*a^2*b^2)*x^2)*sin_integral(d*x))* 
cos(c) + (6*(a*b^3*d*x^4 + 2*a^2*b^2*d*x^3 + a^3*b*d*x^2)*cos_integral((b* 
d*x + a*d)/b) - ((a^2*b^2*d^2 - 12*b^4)*x^4 + 2*(a^3*b*d^2 - 12*a*b^3)*x^3 
 + (a^4*d^2 - 12*a^2*b^2)*x^2)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - 
a*d)/b) - (12*a*b^3*x^3 + 18*a^2*b^2*x^2 + 4*a^3*b*x - a^4)*sin(d*x + c) + 
 (((a^2*b^2*d^2 - 12*b^4)*x^4 + 2*(a^3*b*d^2 - 12*a*b^3)*x^3 + (a^4*d^2 - 
12*a^2*b^2)*x^2)*cos_integral(d*x) - 6*(a*b^3*d*x^4 + 2*a^2*b^2*d*x^3 + a^ 
3*b*d*x^2)*sin_integral(d*x))*sin(c) + (((a^2*b^2*d^2 - 12*b^4)*x^4 + 2*(a 
^3*b*d^2 - 12*a*b^3)*x^3 + (a^4*d^2 - 12*a^2*b^2)*x^2)*cos_integral((b*d*x 
 + a*d)/b) + 6*(a*b^3*d*x^4 + 2*a^2*b^2*d*x^3 + a^3*b*d*x^2)*sin_integral( 
(b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^ 
2)
 

Sympy [F]

\[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x^{3} \left (a + b x\right )^{3}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )}^{3} x^{3}} \,d x } \] Input:

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

Output:

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.55 (sec) , antiderivative size = 24116, normalized size of antiderivative = 63.97 \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(a^2*b^2*d^2*x^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t 
an(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*b^2*d^2*x^4*imag_part(cos_integral(d*x) 
)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*b^2*d^2*x^4*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 
- a^2*b^2*d^2*x^4*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*a^2*b^2*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*ta 
n(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^2*x^4*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 + 2*a^2*b^2*d^2*x^4*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*a^2*b^2*d^2*x^4*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*a^2*b^2*d^2*x^4*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*a^2*b^2*d^2*x^4* 
real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 
2*a^2*b^2*d^2*x^4*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*a^2*b^2*d^2*x^4*real_part(cos_integral(-d*x)) 
*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a^3*b*d^2*x^3*imag_part(co 
s_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2* 
a^3*b*d^2*x^3*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*a^3*b*d^2*x^3*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 - 2*a^3*b*d^2*x^3*imag_part(co...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^3\,{\left (a+b\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sin (c+d x)}{x^3 (a+b x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

( - 12*cos(c + d*x)*a*b**2*x - 12*cos(c + d*x)*b**3*x**2 - 24*int(x/(tan(( 
c + d*x)/2)**2*a**2 + 2*tan((c + d*x)/2)**2*a*b*x + tan((c + d*x)/2)**2*b* 
*2*x**2 + a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**3*d**2*x**2 - 48*int(x/(t 
an((c + d*x)/2)**2*a**2 + 2*tan((c + d*x)/2)**2*a*b*x + tan((c + d*x)/2)** 
2*b**2*x**2 + a**2 + 2*a*b*x + b**2*x**2),x)*a*b**4*d**2*x**3 - 24*int(x/( 
tan((c + d*x)/2)**2*a**2 + 2*tan((c + d*x)/2)**2*a*b*x + tan((c + d*x)/2)* 
*2*b**2*x**2 + a**2 + 2*a*b*x + b**2*x**2),x)*b**5*d**2*x**4 + 2*int(1/(ta 
n((c + d*x)/2)**2*a**2*x**2 + 2*tan((c + d*x)/2)**2*a*b*x**3 + tan((c + d* 
x)/2)**2*b**2*x**4 + a**2*x**2 + 2*a*b*x**3 + b**2*x**4),x)*a**5*d**2*x**2 
 + 4*int(1/(tan((c + d*x)/2)**2*a**2*x**2 + 2*tan((c + d*x)/2)**2*a*b*x**3 
 + tan((c + d*x)/2)**2*b**2*x**4 + a**2*x**2 + 2*a*b*x**3 + b**2*x**4),x)* 
a**4*b*d**2*x**3 + 2*int(1/(tan((c + d*x)/2)**2*a**2*x**2 + 2*tan((c + d*x 
)/2)**2*a*b*x**3 + tan((c + d*x)/2)**2*b**2*x**4 + a**2*x**2 + 2*a*b*x**3 
+ b**2*x**4),x)*a**3*b**2*d**2*x**4 - 24*int(1/(tan((c + d*x)/2)**2*a**2*x 
**2 + 2*tan((c + d*x)/2)**2*a*b*x**3 + tan((c + d*x)/2)**2*b**2*x**4 + a** 
2*x**2 + 2*a*b*x**3 + b**2*x**4),x)*a**3*b**2*x**2 - 48*int(1/(tan((c + d* 
x)/2)**2*a**2*x**2 + 2*tan((c + d*x)/2)**2*a*b*x**3 + tan((c + d*x)/2)**2* 
b**2*x**4 + a**2*x**2 + 2*a*b*x**3 + b**2*x**4),x)*a**2*b**3*x**3 - 24*int 
(1/(tan((c + d*x)/2)**2*a**2*x**2 + 2*tan((c + d*x)/2)**2*a*b*x**3 + tan(( 
c + d*x)/2)**2*b**2*x**4 + a**2*x**2 + 2*a*b*x**3 + b**2*x**4),x)*a*b**...