Integrand size = 17, antiderivative size = 248 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{3 x^2}-\frac {b^2 d \cos (c+d x)}{2 x}+\frac {a^2 d^3 \cos (c+d x)}{24 x}-\frac {1}{3} a b d^3 \cos (c) \operatorname {CosIntegral}(d x)-\frac {1}{2} b^2 d^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {1}{24} a^2 d^4 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {2 a b \sin (c+d x)}{3 x^3}-\frac {b^2 \sin (c+d x)}{2 x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+\frac {a b d^2 \sin (c+d x)}{3 x}-\frac {1}{2} b^2 d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)+\frac {1}{3} a b d^3 \sin (c) \text {Si}(d x) \]
-1/3*a*b*d^3*Ci(d*x)*cos(c)-1/12*a^2*d*cos(d*x+c)/x^3-1/3*a*b*d*cos(d*x+c) /x^2-1/2*b^2*d*cos(d*x+c)/x+1/24*a^2*d^3*cos(d*x+c)/x-1/2*b^2*d^2*cos(c)*S i(d*x)+1/24*a^2*d^4*cos(c)*Si(d*x)-1/2*b^2*d^2*Ci(d*x)*sin(c)+1/24*a^2*d^4 *Ci(d*x)*sin(c)+1/3*a*b*d^3*Si(d*x)*sin(c)-1/4*a^2*sin(d*x+c)/x^4-2/3*a*b* sin(d*x+c)/x^3-1/2*b^2*sin(d*x+c)/x^2+1/24*a^2*d^2*sin(d*x+c)/x^2+1/3*a*b* d^2*sin(d*x+c)/x
Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=\frac {-2 a^2 d x \cos (c+d x)-8 a b d x^2 \cos (c+d x)-12 b^2 d x^3 \cos (c+d x)+a^2 d^3 x^3 \cos (c+d x)+d^2 x^4 \operatorname {CosIntegral}(d x) \left (-8 a b d \cos (c)+\left (-12 b^2+a^2 d^2\right ) \sin (c)\right )-6 a^2 \sin (c+d x)-16 a b x \sin (c+d x)-12 b^2 x^2 \sin (c+d x)+a^2 d^2 x^2 \sin (c+d x)+8 a b d^2 x^3 \sin (c+d x)+d^2 x^4 \left (-12 b^2 \cos (c)+a^2 d^2 \cos (c)+8 a b d \sin (c)\right ) \text {Si}(d x)}{24 x^4} \]
(-2*a^2*d*x*Cos[c + d*x] - 8*a*b*d*x^2*Cos[c + d*x] - 12*b^2*d*x^3*Cos[c + d*x] + a^2*d^3*x^3*Cos[c + d*x] + d^2*x^4*CosIntegral[d*x]*(-8*a*b*d*Cos[ c] + (-12*b^2 + a^2*d^2)*Sin[c]) - 6*a^2*Sin[c + d*x] - 16*a*b*x*Sin[c + d *x] - 12*b^2*x^2*Sin[c + d*x] + a^2*d^2*x^2*Sin[c + d*x] + 8*a*b*d^2*x^3*S in[c + d*x] + d^2*x^4*(-12*b^2*Cos[c] + a^2*d^2*Cos[c] + 8*a*b*d*Sin[c])*S inIntegral[d*x])/(24*x^4)
Time = 0.66 (sec) , antiderivative size = 248, 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.
| \(\displaystyle \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{x^5}+\frac {2 a b \sin (c+d x)}{x^4}+\frac {b^2 \sin (c+d x)}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{24} a^2 d^4 \sin (c) \operatorname {CosIntegral}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)+\frac {a^2 d^3 \cos (c+d x)}{24 x}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {1}{3} a b d^3 \cos (c) \operatorname {CosIntegral}(d x)+\frac {1}{3} a b d^3 \sin (c) \text {Si}(d x)+\frac {a b d^2 \sin (c+d x)}{3 x}-\frac {2 a b \sin (c+d x)}{3 x^3}-\frac {a b d \cos (c+d x)}{3 x^2}-\frac {1}{2} b^2 d^2 \sin (c) \operatorname {CosIntegral}(d x)-\frac {1}{2} b^2 d^2 \cos (c) \text {Si}(d x)-\frac {b^2 \sin (c+d x)}{2 x^2}-\frac {b^2 d \cos (c+d x)}{2 x}\) |
-1/12*(a^2*d*Cos[c + d*x])/x^3 - (a*b*d*Cos[c + d*x])/(3*x^2) - (b^2*d*Cos [c + d*x])/(2*x) + (a^2*d^3*Cos[c + d*x])/(24*x) - (a*b*d^3*Cos[c]*CosInte gral[d*x])/3 - (b^2*d^2*CosIntegral[d*x]*Sin[c])/2 + (a^2*d^4*CosIntegral[ d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4) - (2*a*b*Sin[c + d*x])/(3*x^3 ) - (b^2*Sin[c + d*x])/(2*x^2) + (a^2*d^2*Sin[c + d*x])/(24*x^2) + (a*b*d^ 2*Sin[c + d*x])/(3*x) - (b^2*d^2*Cos[c]*SinIntegral[d*x])/2 + (a^2*d^4*Cos [c]*SinIntegral[d*x])/24 + (a*b*d^3*Sin[c]*SinIntegral[d*x])/3
3.1.17.3.1 Defintions of rubi rules used
Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(d^{4} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )}{d}+\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )}{d^{2}}\right )\) | \(201\) |
| default | \(d^{4} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\operatorname {Si}\left (d x \right ) \sin \left (c \right )}{6}-\frac {\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{6}\right )}{d}+\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )}{d^{2}}\right )\) | \(201\) |
| risch | \(\frac {\cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a b \,d^{3}}{6}-\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a^{2} d^{4}}{48}+\frac {\cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a b \,d^{3}}{6}+\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a^{2} d^{4}}{48}+\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) b^{2} d^{2}}{4}-\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) b^{2} d^{2}}{4}-\frac {i \sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a b \,d^{3}}{6}-\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a^{2} d^{4}}{48}+\frac {i \sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a b \,d^{3}}{6}-\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a^{2} d^{4}}{48}+\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) b^{2} d^{2}}{4}+\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) b^{2} d^{2}}{4}-\frac {i \left (2 i a^{2} d^{9} x^{7}-24 i b^{2} d^{7} x^{7}-16 i a b \,d^{7} x^{6}-4 i a^{2} d^{7} x^{5}\right ) \cos \left (d x +c \right )}{48 d^{6} x^{8}}-\frac {\left (-16 a b \,d^{8} x^{7}-2 a^{2} d^{8} x^{6}+24 b^{2} d^{6} x^{6}+32 a b \,d^{6} x^{5}+12 a^{2} d^{6} x^{4}\right ) \sin \left (d x +c \right )}{48 d^{6} x^{8}}\) | \(339\) |
| meijerg | \(\frac {d^{2} b^{2} \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+\ln \left (d^{2}\right )\right )}{\sqrt {\pi }}+\frac {-6 d^{2} x^{2}+4}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \gamma }{\sqrt {\pi }}+\frac {4 \ln \left (2\right )}{\sqrt {\pi }}+\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \cos \left (d x \right )}{\sqrt {\pi }\, d^{2} x^{2}}+\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}-\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {d^{2} b^{2} \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {4 \cos \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{d^{2} x^{2} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {d^{4} a b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {8 \left (-d^{2} x^{2}+2\right ) d^{2} \cos \left (x \sqrt {d^{2}}\right )}{3 x^{3} \left (d^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 \sin \left (x \sqrt {d^{2}}\right )}{3 d^{2} x^{2} \sqrt {\pi }}+\frac {8 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{3 \sqrt {\pi }}\right )}{8 \sqrt {d^{2}}}+\frac {d^{3} a b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{3 \sqrt {\pi }}+\frac {-\frac {44 d^{2} x^{2}}{9}+8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \cos \left (d x \right )}{3 \sqrt {\pi }\, d^{2} x^{2}}-\frac {16 \left (-\frac {5 d^{2} x^{2}}{2}+5\right ) \sin \left (d x \right )}{15 \sqrt {\pi }\, d^{3} x^{3}}-\frac {8 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{8}+\frac {a^{2} \sqrt {\pi }\, \sin \left (c \right ) d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}+\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {2 \ln \left (d^{2}\right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} d^{4} x^{4}-8 d^{2} x^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {4 \gamma }{3 \sqrt {\pi }}-\frac {4 \ln \left (2\right )}{3 \sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \left (-\frac {15 d^{2} x^{2}}{2}+45\right ) \cos \left (d x \right )}{45 \sqrt {\pi }\, d^{4} x^{4}}+\frac {8 \left (-\frac {15 d^{2} x^{2}}{2}+15\right ) \sin \left (d x \right )}{45 \sqrt {\pi }\, d^{3} x^{3}}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}+\frac {a^{2} \sqrt {\pi }\, \cos \left (c \right ) d^{4} \left (-\frac {8 \left (-\frac {d^{2} x^{2}}{2}+1\right ) \cos \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 \left (-\frac {d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 \,\operatorname {Si}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\) | \(638\) |
d^4*(a^2*(-1/4*sin(d*x+c)/d^4/x^4-1/12*cos(d*x+c)/d^3/x^3+1/24*sin(d*x+c)/ d^2/x^2+1/24*cos(d*x+c)/d/x+1/24*Si(d*x)*cos(c)+1/24*Ci(d*x)*sin(c))+2*a*b /d*(-1/3*sin(d*x+c)/d^3/x^3-1/6*cos(d*x+c)/d^2/x^2+1/6*sin(d*x+c)/d/x+1/6* Si(d*x)*sin(c)-1/6*Ci(d*x)*cos(c))+b^2/d^2*(-1/2*sin(d*x+c)/d^2/x^2-1/2*co s(d*x+c)/d/x-1/2*Si(d*x)*cos(c)-1/2*Ci(d*x)*sin(c)))
Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=-\frac {{\left (8 \, a b d x^{2} + 2 \, a^{2} d x - {\left (a^{2} d^{3} - 12 \, b^{2} d\right )} x^{3}\right )} \cos \left (d x + c\right ) + {\left (8 \, a b d^{3} x^{4} \operatorname {Ci}\left (d x\right ) - {\left (a^{2} d^{4} - 12 \, b^{2} d^{2}\right )} x^{4} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) - {\left (8 \, a b d^{2} x^{3} - 16 \, a b x + {\left (a^{2} d^{2} - 12 \, b^{2}\right )} x^{2} - 6 \, a^{2}\right )} \sin \left (d x + c\right ) - {\left (8 \, a b d^{3} x^{4} \operatorname {Si}\left (d x\right ) + {\left (a^{2} d^{4} - 12 \, b^{2} d^{2}\right )} x^{4} \operatorname {Ci}\left (d x\right )\right )} \sin \left (c\right )}{24 \, x^{4}} \]
-1/24*((8*a*b*d*x^2 + 2*a^2*d*x - (a^2*d^3 - 12*b^2*d)*x^3)*cos(d*x + c) + (8*a*b*d^3*x^4*cos_integral(d*x) - (a^2*d^4 - 12*b^2*d^2)*x^4*sin_integra l(d*x))*cos(c) - (8*a*b*d^2*x^3 - 16*a*b*x + (a^2*d^2 - 12*b^2)*x^2 - 6*a^ 2)*sin(d*x + c) - (8*a*b*d^3*x^4*sin_integral(d*x) + (a^2*d^4 - 12*b^2*d^2 )*x^4*cos_integral(d*x))*sin(c))/x^4
\[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x\right )^{2} \sin {\left (c + d x \right )}}{x^{5}}\, dx \]
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=-\frac {{\left ({\left (a^{2} {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 8 \, {\left (a b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a b {\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{5} - 12 \, {\left (b^{2} {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + b^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 6 \, b^{2} \sin \left (d x + c\right ) + 2 \, {\left (b^{2} d x + 2 \, a b d\right )} \cos \left (d x + c\right )}{2 \, d^{2} x^{4}} \]
-1/2*(((a^2*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a^2*(gamma (-4, I*d*x) + gamma(-4, -I*d*x))*sin(c))*d^6 - 8*(a*b*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*cos(c) + a*b*(-I*gamma(-4, I*d*x) + I*gamma(-4, -I*d*x) )*sin(c))*d^5 - 12*(b^2*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + b^2*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*sin(c))*d^4)*x^4 + 6*b^2*sin( d*x + c) + 2*(b^2*d*x + 2*a*b*d)*cos(d*x + c))/(d^2*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 1712, normalized size of antiderivative = 6.90 \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=\text {Too large to display} \]
-1/48*(a^2*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^ 2 - a^2*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^4* x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^4*x^4 *real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 8*a*b*d^3*x^4*r eal_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 8*a*b*d^3*x^4*re al_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^4*x^4*imag _part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^4*x^4*imag_part(cos_integr al(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 - 16*a*b*d^3*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 16*a*b*d^3*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 3 2*a*b*d^3*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) + a^2*d^4*x^4*im ag_part(cos_integral(d*x))*tan(1/2*c)^2 - a^2*d^4*x^4*imag_part(cos_integr al(-d*x))*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*c)^2 - 12 *b^2*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 12 *b^2*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2 4*b^2*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 8*a*b*d^3*x^ 4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2 + 8*a*b*d^3*x^4*real_part(co s_integral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^4*x^4*real_part(cos_integral(d* x))*tan(1/2*c) - 2*a^2*d^4*x^4*real_part(cos_integral(-d*x))*tan(1/2*c)...
Timed out. \[ \int \frac {(a+b x)^2 \sin (c+d x)}{x^5} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^5} \,d x \]