\(\int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx\) [27]

Optimal result

Integrand size = 17, antiderivative size = 172 \[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\frac {(2 a-b x) \cos (c+d x)}{b^3 d}-\frac {a^3 d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {3 a^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^4}+\frac {\sin (c+d x)}{b^2 d^2}+\frac {a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {a^3 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\frac {-a^2 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac {a d}{b}\right )-3 b \sin \left (c-\frac {a d}{b}\right )\right )+\frac {b \left (b d \left (2 a^2+a b x-b^2 x^2\right ) \cos (c+d x)+\left (a b^2+a^3 d^2+b^3 x\right ) \sin (c+d x)\right )}{d^2 (a+b x)}+a^2 \left (3 b \cos \left (c-\frac {a d}{b}\right )+a d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^5} \] Input:

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

Output:

(-(a^2*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 3*b*Sin[c - (a*d)/ 
b])) + (b*(b*d*(2*a^2 + a*b*x - b^2*x^2)*Cos[c + d*x] + (a*b^2 + a^3*d^2 + 
 b^3*x)*Sin[c + d*x]))/(d^2*(a + b*x)) + a^2*(3*b*Cos[c - (a*d)/b] + a*d*S 
in[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^5
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.05, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*a*Cos[c + d*x])/(b^3*d) - (x*Cos[c + d*x])/(b^2*d) - (a^3*d*Cos[c - (a* 
d)/b]*CosIntegral[(a*d)/b + d*x])/b^5 + (3*a^2*CosIntegral[(a*d)/b + d*x]* 
Sin[c - (a*d)/b])/b^4 + Sin[c + d*x]/(b^2*d^2) + (a^3*Sin[c + d*x])/(b^4*( 
a + b*x)) + (3*a^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4 + (a^3 
*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 1.20 (sec) , antiderivative size = 607, normalized size of antiderivative = 3.53

method	result	size
risch	\(-\frac {i \left (-2 i b^{4} d^{3} x^{4}+4 i a \,b^{3} d^{3} x^{3}-6 i b^{4} c \,d^{2} x^{3}+6 i a^{2} b^{2} d^{3} x^{2}-8 i a^{3} b \,d^{3} x +18 i a^{2} b^{2} c \,d^{2} x -8 i a^{4} d^{3}+12 i a^{3} b c \,d^{2}\right ) \cos \left (d x +c \right )}{2 d^{2} b^{3} \left (b x +a \right ) \left (-d x b +2 a d -3 b c \right ) \left (-d x b -a d \right )}+\frac {\left (2 a^{3} b^{2} d^{4} x^{2}-2 a^{4} b \,d^{4} x +6 a^{3} b^{2} c \,d^{3} x +2 b^{5} d^{2} x^{3}-4 a^{5} d^{4}+6 a^{4} b c \,d^{3}+6 b^{5} c d \,x^{2}-6 a^{2} b^{3} d^{2} x +12 a \,b^{4} c d x -4 a^{3} b^{2} d^{2}+6 a^{2} b^{3} c d \right ) \sin \left (d x +c \right )}{2 d^{2} b^{4} \left (b x +a \right ) \left (-d x b +2 a d -3 b c \right ) \left (-d x b -a d \right )}+\frac {d \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a^{3}}{2 b^{5}}+\frac {d \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a^{3}}{2 b^{5}}-\frac {3 i \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a^{2}}{2 b^{4}}+\frac {3 i \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a^{2}}{2 b^{4}}+\frac {i d \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a^{3}}{2 b^{5}}-\frac {i d \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a^{3}}{2 b^{5}}+\frac {3 \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a^{2}}{2 b^{4}}+\frac {3 \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a^{2}}{2 b^{4}}\)	\(607\)
derivativedivides	\(\frac {-d^{2} c^{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 )+\frac {3 d^{2} c^{2} \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 )}{b}-\frac {3 d^{2} c^{2} \left (a d -b c \right ) \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 )}{b}+\frac {6 \left (a d -b c \right ) d^{2} c \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 )}{b^{2}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{2} c \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 )}{b^{2}}+\frac {3 d^{2} c \cos \left (d x +c \right )}{b^{2}}+\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{2} \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 )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{2} \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 )}{b^{3}}-\frac {d^{2} \left (2 a d -2 b c -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{3}}}{d^{4}}\)	\(851\)
default	\(\frac {-d^{2} c^{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 )+\frac {3 d^{2} c^{2} \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 )}{b}-\frac {3 d^{2} c^{2} \left (a d -b c \right ) \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 )}{b}+\frac {6 \left (a d -b c \right ) d^{2} c \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 )}{b^{2}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{2} c \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 )}{b^{2}}+\frac {3 d^{2} c \cos \left (d x +c \right )}{b^{2}}+\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{2} \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 )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{2} \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 )}{b^{3}}-\frac {d^{2} \left (2 a d -2 b c -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{3}}}{d^{4}}\)	\(851\)

Input:

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

Output:

-1/2*I/d^2/b^3*(-2*I*b^4*d^3*x^4+18*I*a^2*b^2*c*d^2*x+4*I*a*b^3*d^3*x^3-6* 
I*b^4*c*d^2*x^3+12*I*a^3*b*c*d^2+6*I*a^2*b^2*d^3*x^2-8*I*a^3*b*d^3*x-8*I*a 
^4*d^3)/(b*x+a)/(-b*d*x+2*a*d-3*b*c)/(-b*d*x-a*d)*cos(d*x+c)+1/2/d^2/b^4*( 
2*a^3*b^2*d^4*x^2-2*a^4*b*d^4*x+6*a^3*b^2*c*d^3*x+2*b^5*d^2*x^3-4*a^5*d^4+ 
6*a^4*b*c*d^3+6*b^5*c*d*x^2-6*a^2*b^3*d^2*x+12*a*b^4*c*d*x-4*a^3*b^2*d^2+6 
*a^2*b^3*c*d)/(b*x+a)/(-b*d*x+2*a*d-3*b*c)/(-b*d*x-a*d)*sin(d*x+c)+1/2*d/b 
^5*cos((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)*a^3+1/2*d/b^5*cos((a*d-b*c)/b)*Ei( 
1,-I*(b*x+a)*d/b)*a^3-3/2*I/b^4*cos((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)*a^2+3 
/2*I/b^4*cos((a*d-b*c)/b)*Ei(1,-I*(b*x+a)*d/b)*a^2+1/2*I*d/b^5*sin((a*d-b* 
c)/b)*Ei(1,I*(b*x+a)*d/b)*a^3-1/2*I*d/b^5*sin((a*d-b*c)/b)*Ei(1,-I*(b*x+a) 
*d/b)*a^3+3/2/b^4*sin((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)*a^2+3/2/b^4*sin((a* 
d-b*c)/b)*Ei(1,-I*(b*x+a)*d/b)*a^2
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.43 \[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=-\frac {{\left (b^{4} d x^{2} - a b^{3} d x - 2 \, a^{2} b^{2} d\right )} \cos \left (d x + c\right ) + {\left ({\left (a^{3} b d^{3} x + a^{4} d^{3}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - 3 \, {\left (a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left (a^{3} b d^{2} + b^{4} x + a b^{3}\right )} \sin \left (d x + c\right ) + {\left (3 \, {\left (a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{3} b d^{3} x + a^{4} d^{3}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{b^{6} d^{2} x + a b^{5} d^{2}} \] Input:

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

Output:

-((b^4*d*x^2 - a*b^3*d*x - 2*a^2*b^2*d)*cos(d*x + c) + ((a^3*b*d^3*x + a^4 
*d^3)*cos_integral((b*d*x + a*d)/b) - 3*(a^2*b^2*d^2*x + a^3*b*d^2)*sin_in 
tegral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) - (a^3*b*d^2 + b^4*x + a*b^3) 
*sin(d*x + c) + (3*(a^2*b^2*d^2*x + a^3*b*d^2)*cos_integral((b*d*x + a*d)/ 
b) + (a^3*b*d^3*x + a^4*d^3)*sin_integral((b*d*x + a*d)/b))*sin(-(b*c - a* 
d)/b))/(b^6*d^2*x + a*b^5*d^2)
 

Sympy [F]

\[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^{3} \sin {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x + a\right )}^{2}} \,d x } \] Input:

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

Output:

-1/2*((b^2*cos(c)^2 + b^2*sin(c)^2)*d*x^3*cos(d*x + c) - 2*((a^2*(I*exp_in 
tegral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/ 
b))*cos(c)^2 + a^2*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integ 
ral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b) - (a^2*(exp_ 
integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/ 
b))*cos(c)^2 + a^2*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_ 
e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 
- 2*((a^2*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, 
-(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*(I*exp_integral_e(3, (I*b*d*x + I*a* 
d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a 
*d)/b) - (a^2*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, 
-(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*(exp_integral_e(3, (I*b*d*x + I*a*d) 
/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/ 
b))*sin(d*x + c)^2 + ((b^2*d*x^3*cos(c) + b^2*x^2*sin(c) + 2*a*b*x*sin(c)) 
*cos(d*x + c)^2 + (b^2*d*x^3*cos(c) + b^2*x^2*sin(c) + 2*a*b*x*sin(c))*sin 
(d*x + c)^2)*cos(d*x + 2*c) + 2*(((a^2*b^4*cos(c)^2 + a^2*b^4*sin(c)^2)*d^ 
3*x^2 + 2*(a^3*b^3*cos(c)^2 + a^3*b^3*sin(c)^2)*d^3*x + (a^4*b^2*cos(c)^2 
+ a^4*b^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a^2*b^4*cos(c)^2 + a^2*b^4*sin 
(c)^2)*d^3*x^2 + 2*(a^3*b^3*cos(c)^2 + a^3*b^3*sin(c)^2)*d^3*x + (a^4*b^2* 
cos(c)^2 + a^4*b^2*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(x*cos(d*x +...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1474 vs. \(2 (177) = 354\).

Time = 0.21 (sec) , antiderivative size = 1474, normalized size of antiderivative = 8.57 \[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-((b*x + a)*a^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*cos(-(b*c - a*d)/b 
)*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d) 
/b) - a^3*b*c*d^3*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + 
a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^4*d^4*cos(-(b*c - a*d)/b)*cos_ 
integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 
(b*x + a)*a^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*sin(-(b*c - a*d)/b)* 
sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b 
) - a^3*b*c*d^3*sin(-(b*c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) 
 - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^4*d^4*sin(-(b*c - a*d)/b)*sin_in 
tegral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 3* 
(b*x + a)*a^2*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*cos_integral(((b*x 
 + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d) 
/b) - 3*a^2*b^2*c*d^2*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + 
a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) + 3*a^3*b*d^3*cos_integral(((b 
*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a* 
d)/b) - 3*(b*x + a)*a^2*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*cos(-(b* 
c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - 
b*c + a*d)/b) + 3*a^2*b^2*c*d^2*cos(-(b*c - a*d)/b)*sin_integral(((b*x + a 
)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - 3*a^3*b*d^3*cos(-( 
b*c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^3\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {\sin \left (d x +c \right ) x^{3}}{b^{2} x^{2}+2 a b x +a^{2}}d x \] Input:

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

Output:

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