Integrand size = 19, antiderivative size = 167 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=-\frac {a^2 d \cos (c+d x)}{12 x^3}+\frac {a^2 d^3 \cos (c+d x)}{24 x}-\frac {b^2 x \cos (c+d x)}{d}+2 a b d \cos (c) \operatorname {CosIntegral}(d x)+\frac {1}{24} a^2 d^4 \operatorname {CosIntegral}(d x) \sin (c)+\frac {b^2 \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{4 x^4}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}-\frac {2 a b \sin (c+d x)}{x}+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)-2 a b d \sin (c) \text {Si}(d x) \] Output:
-1/12*a^2*d*cos(d*x+c)/x^3+1/24*a^2*d^3*cos(d*x+c)/x-b^2*x*cos(d*x+c)/d+2* a*b*d*cos(c)*Ci(d*x)+1/24*a^2*d^4*Ci(d*x)*sin(c)+b^2*sin(d*x+c)/d^2-1/4*a^ 2*sin(d*x+c)/x^4+1/24*a^2*d^2*sin(d*x+c)/x^2-2*a*b*sin(d*x+c)/x+1/24*a^2*d ^4*cos(c)*Si(d*x)-2*a*b*d*sin(c)*Si(d*x)
Time = 0.59 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\frac {1}{24} \left (-\frac {2 a^2 d \cos (c+d x)}{x^3}+\frac {a^2 d^3 \cos (c+d x)}{x}-\frac {24 b^2 x \cos (c+d x)}{d}+a d \operatorname {CosIntegral}(d x) \left (48 b \cos (c)+a d^3 \sin (c)\right )+\frac {24 b^2 \sin (c+d x)}{d^2}-\frac {6 a^2 \sin (c+d x)}{x^4}+\frac {a^2 d^2 \sin (c+d x)}{x^2}-\frac {48 a b \sin (c+d x)}{x}+a d \left (a d^3 \cos (c)-48 b \sin (c)\right ) \text {Si}(d x)\right ) \] Input:
Integrate[((a + b*x^3)^2*Sin[c + d*x])/x^5,x]
Output:
((-2*a^2*d*Cos[c + d*x])/x^3 + (a^2*d^3*Cos[c + d*x])/x - (24*b^2*x*Cos[c + d*x])/d + a*d*CosIntegral[d*x]*(48*b*Cos[c] + a*d^3*Sin[c]) + (24*b^2*Si n[c + d*x])/d^2 - (6*a^2*Sin[c + d*x])/x^4 + (a^2*d^2*Sin[c + d*x])/x^2 - (48*a*b*Sin[c + d*x])/x + a*d*(a*d^3*Cos[c] - 48*b*Sin[c])*SinIntegral[d*x ])/24
Time = 0.50 (sec) , antiderivative size = 167, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{x^5}+\frac {2 a b \sin (c+d x)}{x^2}+b^2 x \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{24} a^2 d^4 \sin (c) \operatorname {CosIntegral}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)+\frac {a^2 d^3 \cos (c+d x)}{24 x}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a^2 d \cos (c+d x)}{12 x^3}+2 a b d \cos (c) \operatorname {CosIntegral}(d x)-2 a b d \sin (c) \text {Si}(d x)-\frac {2 a b \sin (c+d x)}{x}+\frac {b^2 \sin (c+d x)}{d^2}-\frac {b^2 x \cos (c+d x)}{d}\) |
Input:
Int[((a + b*x^3)^2*Sin[c + d*x])/x^5,x]
Output:
-1/12*(a^2*d*Cos[c + d*x])/x^3 + (a^2*d^3*Cos[c + d*x])/(24*x) - (b^2*x*Co s[c + d*x])/d + 2*a*b*d*Cos[c]*CosIntegral[d*x] + (a^2*d^4*CosIntegral[d*x ]*Sin[c])/24 + (b^2*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/(4*x^4) + (a^2* d^2*Sin[c + d*x])/(24*x^2) - (2*a*b*Sin[c + d*x])/x + (a^2*d^4*Cos[c]*SinI ntegral[d*x])/24 - 2*a*b*d*Sin[c]*SinIntegral[d*x]
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 = 1.72 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(d^{4} \left (\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d^{3}}+\frac {6 b^{2} c \cos \left (d x +c \right )}{d^{6}}+\frac {\left (5 c +1\right ) b^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}+a^{2} \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )\right )\) | \(167\) |
| default | \(d^{4} \left (\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )}{d^{3}}+\frac {6 b^{2} c \cos \left (d x +c \right )}{d^{6}}+\frac {\left (5 c +1\right ) b^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}+a^{2} \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{24}+\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{24}\right )\right )\) | \(167\) |
| risch | \(\frac {-\cos \left (c \right ) \pi \,\operatorname {csgn}\left (d x \right ) a^{2} d^{6} x^{4}-96 i \cos \left (c \right ) \operatorname {Si}\left (d x \right ) a b \,d^{3} x^{4}+2 \cos \left (c \right ) \operatorname {Si}\left (d x \right ) a^{2} d^{6} x^{4}+i \sin \left (c \right ) \pi \,\operatorname {csgn}\left (d x \right ) a^{2} d^{6} x^{4}+48 \sin \left (c \right ) \pi \,\operatorname {csgn}\left (d x \right ) a b \,d^{3} x^{4}-2 \sin \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) a^{2} d^{6} x^{4}-96 \sin \left (c \right ) \operatorname {Si}\left (d x \right ) a b \,d^{3} x^{4}-2 i \sin \left (c \right ) \operatorname {Si}\left (d x \right ) a^{2} d^{6} x^{4}+48 i \cos \left (c \right ) \pi \,\operatorname {csgn}\left (d x \right ) a b \,d^{3} x^{4}+2 \cos \left (d x +c \right ) a^{2} d^{5} x^{3}-96 \cos \left (c \right ) \operatorname {expIntegral}_{1}\left (-i d x \right ) a b \,d^{3} x^{4}+2 \sin \left (d x +c \right ) a^{2} d^{4} x^{2}-48 \cos \left (d x +c \right ) b^{2} d \,x^{5}-96 \sin \left (d x +c \right ) a b \,d^{2} x^{3}+48 b^{2} x^{4} \sin \left (d x +c \right )-4 \cos \left (d x +c \right ) a^{2} d^{3} x -12 \sin \left (d x +c \right ) a^{2} d^{2}}{48 d^{2} x^{4}}\) | \(297\) |
| meijerg | \(\frac {2 b^{2} \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {x d \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {2 b^{2} \cos \left (c \right ) \sqrt {\pi }\, \left (-\frac {d x \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {d^{2} a b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{2 \sqrt {d^{2}}}+\frac {d a b \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\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 \sin \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {a^{2} \sin \left (c \right ) \sqrt {\pi }\, 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} x^{4} d^{4}-8 x^{2} d^{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 x^{2} d^{2}}{2}+45\right ) \cos \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}+\frac {8 \left (-\frac {15 x^{2} d^{2}}{2}+15\right ) \sin \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}+\frac {a^{2} \cos \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 \left (-\frac {x^{2} d^{2}}{2}+1\right ) \cos \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 \left (-\frac {x^{2} d^{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}\) | \(449\) |
Input:
int((b*x^3+a)^2*sin(d*x+c)/x^5,x,method=_RETURNVERBOSE)
Output:
d^4*(2/d^3*a*b*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+6/d^6*b^2*c *cos(d*x+c)+(5*c+1)/d^6*b^2*(sin(d*x+c)-cos(d*x+c)*(d*x+c))+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)))
Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\frac {{\left (a^{2} d^{5} x^{3} - 24 \, b^{2} d x^{5} - 2 \, a^{2} d^{3} x\right )} \cos \left (d x + c\right ) + {\left (a^{2} d^{6} x^{4} \operatorname {Si}\left (d x\right ) + 48 \, a b d^{3} x^{4} \operatorname {Ci}\left (d x\right )\right )} \cos \left (c\right ) + {\left (a^{2} d^{4} x^{2} - 48 \, a b d^{2} x^{3} + 24 \, b^{2} x^{4} - 6 \, a^{2} d^{2}\right )} \sin \left (d x + c\right ) + {\left (a^{2} d^{6} x^{4} \operatorname {Ci}\left (d x\right ) - 48 \, a b d^{3} x^{4} \operatorname {Si}\left (d x\right )\right )} \sin \left (c\right )}{24 \, d^{2} x^{4}} \] Input:
integrate((b*x^3+a)^2*sin(d*x+c)/x^5,x, algorithm="fricas")
Output:
1/24*((a^2*d^5*x^3 - 24*b^2*d*x^5 - 2*a^2*d^3*x)*cos(d*x + c) + (a^2*d^6*x ^4*sin_integral(d*x) + 48*a*b*d^3*x^4*cos_integral(d*x))*cos(c) + (a^2*d^4 *x^2 - 48*a*b*d^2*x^3 + 24*b^2*x^4 - 6*a^2*d^2)*sin(d*x + c) + (a^2*d^6*x^ 4*cos_integral(d*x) - 48*a*b*d^3*x^4*sin_integral(d*x))*sin(c))/(d^2*x^4)
\[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x^{3}\right )^{2} \sin {\left (c + d x \right )}}{x^{5}}\, dx \] Input:
integrate((b*x**3+a)**2*sin(d*x+c)/x**5,x)
Output:
Integral((a + b*x**3)**2*sin(c + d*x)/x**5, x)
Result contains complex when optimal does not.
Time = 6.41 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\frac {{\left ({\left (a^{2} {\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - a^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{7} - 48 \, {\left (a b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a b {\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} - 2 \, {\left (b^{2} d^{2} x^{5} + 2 \, a b d^{2} x^{2} - 12 \, a b\right )} \cos \left (d x + c\right ) + 2 \, {\left (b^{2} d x^{4} - 4 \, a b d x\right )} \sin \left (d x + c\right )}{2 \, d^{3} x^{4}} \] Input:
integrate((b*x^3+a)^2*sin(d*x+c)/x^5,x, algorithm="maxima")
Output:
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^7 - 48*(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^4)*x^4 - 2*(b^2*d^2*x^5 + 2*a*b*d^2*x^2 - 12*a*b)*cos(d*x + c ) + 2*(b^2*d*x^4 - 4*a*b*d*x)*sin(d*x + c))/(d^3*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1255, normalized size of antiderivative = 7.51 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\text {Too large to display} \] Input:
integrate((b*x^3+a)^2*sin(d*x+c)/x^5,x, algorithm="giac")
Output:
-1/48*(a^2*d^6*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^ 2 - a^2*d^6*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^6*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^6* x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^6*x^4 *real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - a^2*d^6*x^4*ima g_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^6*x^4*imag_part(cos_integ ral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^6*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 + a^2*d^6*x^4*imag_part(cos_integral(d*x))*tan(1/2*c)^2 - a^2*d^6*x^4*ima g_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a^2*d^6*x^4*sin_integral(d*x)* tan(1/2*c)^2 - 2*a^2*d^6*x^4*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a ^2*d^6*x^4*real_part(cos_integral(-d*x))*tan(1/2*c) - 2*a^2*d^5*x^3*tan(1/ 2*d*x)^2*tan(1/2*c)^2 + 48*a*b*d^3*x^4*real_part(cos_integral(d*x))*tan(1/ 2*d*x)^2*tan(1/2*c)^2 + 48*a*b*d^3*x^4*real_part(cos_integral(-d*x))*tan(1 /2*d*x)^2*tan(1/2*c)^2 - a^2*d^6*x^4*imag_part(cos_integral(d*x)) + a^2*d^ 6*x^4*imag_part(cos_integral(-d*x)) - 2*a^2*d^6*x^4*sin_integral(d*x) + 96 *a*b*d^3*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 96*a *b*d^3*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 192*a *b*d^3*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^5*x^3*tan (1/2*d*x)^2 - 48*a*b*d^3*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2 - 48*a*b*d^3*x^4*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 + 8*a^2*d^...
Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^5} \,d x \] Input:
int((sin(c + d*x)*(a + b*x^3)^2)/x^5,x)
Output:
int((sin(c + d*x)*(a + b*x^3)^2)/x^5, x)
\[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx=\frac {-24 \cos \left (d x +c \right ) a b \,d^{2} x^{2}+144 \cos \left (d x +c \right ) a b -12 \cos \left (d x +c \right ) b^{2} d^{2} x^{5}-6 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{4}+x^{4}}d x \right ) a^{2} d^{4} x^{4}+1152 \left (\int \frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{5}+x^{5}}d x \right ) a b \,x^{4}-3 \sin \left (d x +c \right ) a^{2} d^{3}-48 \sin \left (d x +c \right ) a b d x +12 \sin \left (d x +c \right ) b^{2} d \,x^{4}-a^{2} d^{4} x +144 a b -12 b^{2} c d \,x^{4}}{12 d^{3} x^{4}} \] Input:
int((b*x^3+a)^2*sin(d*x+c)/x^5,x)
Output:
( - 24*cos(c + d*x)*a*b*d**2*x**2 + 144*cos(c + d*x)*a*b - 12*cos(c + d*x) *b**2*d**2*x**5 - 6*int(tan((c + d*x)/2)**2/(tan((c + d*x)/2)**2*x**4 + x* *4),x)*a**2*d**4*x**4 + 1152*int(1/(tan((c + d*x)/2)**2*x**5 + x**5),x)*a* b*x**4 - 3*sin(c + d*x)*a**2*d**3 - 48*sin(c + d*x)*a*b*d*x + 12*sin(c + d *x)*b**2*d*x**4 - a**2*d**4*x + 144*a*b - 12*b**2*c*d*x**4)/(12*d**3*x**4)