Integrand size = 19, antiderivative size = 177 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+\frac {a^2 d^3 \cos (c+d x)}{24 x}+b^2 \operatorname {CosIntegral}(d x) \sin (c)-a b d^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {1}{24} a^2 d^4 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text {Si}(d x)-a b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x) \]
-1/12*a^2*d*cos(d*x+c)/x^3-a*b*d*cos(d*x+c)/x+1/24*a^2*d^3*cos(d*x+c)/x+b^ 2*cos(c)*Si(d*x)-a*b*d^2*cos(c)*Si(d*x)+1/24*a^2*d^4*cos(c)*Si(d*x)+b^2*Ci (d*x)*sin(c)-a*b*d^2*Ci(d*x)*sin(c)+1/24*a^2*d^4*Ci(d*x)*sin(c)-1/4*a^2*si n(d*x+c)/x^4-a*b*sin(d*x+c)/x^2+1/24*a^2*d^2*sin(d*x+c)/x^2
Time = 0.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=\frac {a d x \left (-24 b x^2+a \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+\left (24 b^2-24 a b d^2+a^2 d^4\right ) x^4 \operatorname {CosIntegral}(d x) \sin (c)+a \left (-24 b x^2+a \left (-6+d^2 x^2\right )\right ) \sin (c+d x)+\left (24 b^2-24 a b d^2+a^2 d^4\right ) x^4 \cos (c) \text {Si}(d x)}{24 x^4} \]
(a*d*x*(-24*b*x^2 + a*(-2 + d^2*x^2))*Cos[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*CosIntegral[d*x]*Sin[c] + a*(-24*b*x^2 + a*(-6 + d^2*x^2))* Sin[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*Cos[c]*SinIntegral[d*x] )/(24*x^4)
Time = 0.52 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3820, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{x^5}+\frac {2 a b \sin (c+d x)}{x^3}+\frac {b^2 \sin (c+d x)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{24} a^2 d^4 \sin (c) \operatorname {CosIntegral}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)+\frac {a^2 d^3 \cos (c+d x)}{24 x}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a^2 d \cos (c+d x)}{12 x^3}-a b d^2 \sin (c) \operatorname {CosIntegral}(d x)-a b d^2 \cos (c) \text {Si}(d x)-\frac {a b \sin (c+d x)}{x^2}-\frac {a b d \cos (c+d x)}{x}+b^2 \sin (c) \operatorname {CosIntegral}(d x)+b^2 \cos (c) \text {Si}(d x)\) |
-1/12*(a^2*d*Cos[c + d*x])/x^3 - (a*b*d*Cos[c + d*x])/x + (a^2*d^3*Cos[c + d*x])/(24*x) + b^2*CosIntegral[d*x]*Sin[c] - a*b*d^2*CosIntegral[d*x]*Sin [c] + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4) - (a*b*Sin[c + d*x])/x^2 + (a^2*d^2*Sin[c + d*x])/(24*x^2) + b^2*Cos[c]*SinI ntegral[d*x] - a*b*d^2*Cos[c]*SinIntegral[d*x] + (a^2*d^4*Cos[c]*SinIntegr al[d*x])/24
3.1.56.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_ )], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x ], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.42 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(d^{4} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{4}}\right )\) | \(157\) |
| default | \(d^{4} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{d^{4}}\right )\) | \(157\) |
| risch | \(\frac {i \operatorname {Ei}_{1}\left (-i d x \right ) \cos \left (c \right ) a^{2} d^{4}}{48}-\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a^{2} d^{4}}{48}-\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a b \,d^{2}}{2}+\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a b \,d^{2}}{2}+\frac {i \operatorname {Ei}_{1}\left (-i d x \right ) \cos \left (c \right ) b^{2}}{2}-\frac {i \cos \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) b^{2}}{2}-\frac {\operatorname {Ei}_{1}\left (-i d x \right ) \sin \left (c \right ) a^{2} d^{4}}{48}-\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a^{2} d^{4}}{48}+\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (-i d x \right ) a b \,d^{2}}{2}+\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) a b \,d^{2}}{2}-\frac {\operatorname {Ei}_{1}\left (-i d x \right ) \sin \left (c \right ) b^{2}}{2}-\frac {\sin \left (c \right ) \operatorname {Ei}_{1}\left (i d x \right ) b^{2}}{2}-\frac {i \left (2 i a^{2} d^{11} x^{7}-48 i a b \,d^{9} x^{7}-4 i a^{2} d^{9} x^{5}\right ) \cos \left (d x +c \right )}{48 d^{8} x^{8}}-\frac {\left (-2 a^{2} d^{10} x^{6}+48 a b \,d^{8} x^{6}+12 a^{2} d^{8} x^{4}\right ) \sin \left (d x +c \right )}{48 d^{8} x^{8}}\) | \(294\) |
| meijerg | \(\frac {b^{2} \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+b^{2} \cos \left (c \right ) \operatorname {Si}\left (d x \right )+\frac {d^{2} a b \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 )}{4}+\frac {d^{2} a b \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 )}{4}+\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}\) | \(486\) |
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/d^2 *a*b*(-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))+1/d^4*b^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c)))
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=\frac {{\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + {\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - {\left (2 \, a^{2} d x - {\left (a^{2} d^{3} - 24 \, a b d\right )} x^{3}\right )} \cos \left (d x + c\right ) + {\left ({\left (a^{2} d^{2} - 24 \, a b\right )} x^{2} - 6 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, x^{4}} \]
1/24*((a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos_integral(d*x)*sin(c) + (a^2* d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos(c)*sin_integral(d*x) - (2*a^2*d*x - (a^ 2*d^3 - 24*a*b*d)*x^3)*cos(d*x + c) + ((a^2*d^2 - 24*a*b)*x^2 - 6*a^2)*sin (d*x + c))/x^4
\[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x^{2}\right )^{2} \sin {\left (c + d x \right )}}{x^{5}}\, dx \]
Result contains complex when optimal does not.
Time = 8.17 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=-\frac {{\left ({\left (a^{2} {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{8} - 24 \, {\left (a b {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 24 \, {\left (b^{2} {\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - b^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} d^{3} x^{3} + 2 \, {\left (a b d^{3} - b^{2} d\right )} x\right )} \cos \left (d x + c\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 6 \, a b d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{4} x^{4}} \]
-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^8 - 24*(a*b*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a*b*(gamma(-4, I*d*x) + gamma(-4, -I*d*x) )*sin(c))*d^6 - 24*(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 + 2*(b^2*d^ 3*x^3 + 2*(a*b*d^3 - b^2*d)*x)*cos(d*x + c) + 2*(b^2*d^2*x^2 + 6*a*b*d^2 - 6*b^2)*sin(d*x + c))/(d^4*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.30 (sec) , antiderivative size = 1497, normalized size of antiderivative = 8.46 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=\text {Too large to display} \]
-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) - a^2*d^4*x^4*ima g_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^4*x^4*imag_part(cos_integ ral(-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 + a^2*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*c)^2 - a^2*d^4*x^4*ima g_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral(d*x)* tan(1/2*c)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2* tan(1/2*c)^2 + 24*a*b*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 *tan(1/2*c)^2 - 48*a*b*d^2*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*c) - 2*a^2*d^4*x^4 *real_part(cos_integral(-d*x))*tan(1/2*c) + 48*a*b*d^2*x^4*real_part(cos_i ntegral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 48*a*b*d^2*x^4*real_part(cos_int egral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^3*x^3*tan(1/2*d*x)^2*tan( 1/2*c)^2 - a^2*d^4*x^4*imag_part(cos_integral(d*x)) + a^2*d^4*x^4*imag_par t(cos_integral(-d*x)) - 2*a^2*d^4*x^4*sin_integral(d*x) + 24*a*b*d^2*x^4*i mag_part(cos_integral(d*x))*tan(1/2*d*x)^2 - 24*a*b*d^2*x^4*imag_part(cos_ integral(-d*x))*tan(1/2*d*x)^2 + 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1...
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^5} \,d x \]