Integrand size = 17, antiderivative size = 70 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=-\frac {b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{2 x}-\frac {1}{2} a d^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x) \]
-b*cos(d*x+c)/d-1/2*a*d*cos(d*x+c)/x-1/2*a*d^2*cos(c)*Si(d*x)-1/2*a*d^2*Ci (d*x)*sin(c)-1/2*a*sin(d*x+c)/x^2
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\frac {1}{2} \left (-\frac {2 b \cos (c+d x)}{d}-\frac {a d \cos (c+d x)}{x}-a d^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {a \sin (c+d x)}{x^2}-a d^2 \cos (c) \text {Si}(d x)\right ) \]
((-2*b*Cos[c + d*x])/d - (a*d*Cos[c + d*x])/x - a*d^2*CosIntegral[d*x]*Sin [c] - (a*Sin[c + d*x])/x^2 - a*d^2*Cos[c]*SinIntegral[d*x])/2
Time = 0.28 (sec) , antiderivative size = 70, 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 = {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.
| \(\displaystyle \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a \sin (c+d x)}{x^3}+b \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} a d^2 \sin (c) \operatorname {CosIntegral}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {a d \cos (c+d x)}{2 x}-\frac {b \cos (c+d x)}{d}\) |
-((b*Cos[c + d*x])/d) - (a*d*Cos[c + d*x])/(2*x) - (a*d^2*CosIntegral[d*x] *Sin[c])/2 - (a*Sin[c + d*x])/(2*x^2) - (a*d^2*Cos[c]*SinIntegral[d*x])/2
3.1.85.3.1 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 = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(d^{2} \left (a \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )-\frac {b \cos \left (d x +c \right )}{d^{3}}\right )\) | \(65\) |
| default | \(d^{2} \left (a \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}\right )-\frac {b \cos \left (d x +c \right )}{d^{3}}\right )\) | \(65\) |
| risch | \(-\frac {i d^{2} \cos \left (c \right ) a \,\operatorname {Ei}_{1}\left (-i d x \right )}{4}+\frac {i d^{2} \cos \left (c \right ) a \,\operatorname {Ei}_{1}\left (i d x \right )}{4}+\frac {d^{2} \sin \left (c \right ) a \,\operatorname {Ei}_{1}\left (-i d x \right )}{4}+\frac {d^{2} \sin \left (c \right ) a \,\operatorname {Ei}_{1}\left (i d x \right )}{4}-\frac {i \left (-2 i a \,d^{6} x^{3}-4 i b \,d^{4} x^{4}\right ) \cos \left (d x +c \right )}{4 d^{5} x^{4}}-\frac {a \sin \left (d x +c \right )}{2 x^{2}}\) | \(112\) |
| meijerg | \(\frac {b \sin \left (c \right ) \sin \left (d x \right )}{d}+\frac {b \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{2} \left (-\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+\ln \left (d^{2}\right )\right )}{\sqrt {\pi }}+\frac {-6 d^{2} x^{2}+4}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \gamma }{\sqrt {\pi }}+\frac {4 \ln \left (2\right )}{\sqrt {\pi }}+\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \cos \left (d x \right )}{\sqrt {\pi }\, d^{2} x^{2}}+\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}-\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d^{2} \left (-\frac {4 \cos \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{d^{2} x^{2} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\) | \(211\) |
d^2*(a*(-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))-b*cos(d*x+c)/d^3)
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=-\frac {a d^{3} x^{2} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + a d^{3} x^{2} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) + a d \sin \left (d x + c\right ) + {\left (a d^{2} x + 2 \, b x^{2}\right )} \cos \left (d x + c\right )}{2 \, d x^{2}} \]
-1/2*(a*d^3*x^2*cos_integral(d*x)*sin(c) + a*d^3*x^2*cos(c)*sin_integral(d *x) + a*d*sin(d*x + c) + (a*d^2*x + 2*b*x^2)*cos(d*x + c))/(d*x^2)
\[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x^{3}\right ) \sin {\left (c + d x \right )}}{x^{3}}\, dx \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 1146, normalized size of antiderivative = 16.37 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\text {Too large to display} \]
1/4*(((I*exp_integral_e(3, I*d*x) - I*exp_integral_e(3, -I*d*x))*cos(c)^3 + (I*exp_integral_e(3, I*d*x) - I*exp_integral_e(3, -I*d*x))*cos(c)*sin(c) ^2 + (exp_integral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*sin(c)^3 + (I* exp_integral_e(3, I*d*x) - I*exp_integral_e(3, -I*d*x))*cos(c) + ((exp_int egral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*cos(c)^2 + exp_integral_e(3 , I*d*x) + exp_integral_e(3, -I*d*x))*sin(c))*b*c^3/((d*x + c)^2*(cos(c)^2 + sin(c)^2)*d^3 - 2*(c*cos(c)^2 + c*sin(c)^2)*(d*x + c)*d^3 + (c^2*cos(c) ^2 + c^2*sin(c)^2)*d^3) - ((I*exp_integral_e(3, I*d*x) - I*exp_integral_e( 3, -I*d*x))*cos(c)^3 + (I*exp_integral_e(3, I*d*x) - I*exp_integral_e(3, - I*d*x))*cos(c)*sin(c)^2 + (exp_integral_e(3, I*d*x) + exp_integral_e(3, -I *d*x))*sin(c)^3 + (I*exp_integral_e(3, I*d*x) - I*exp_integral_e(3, -I*d*x ))*cos(c) + ((exp_integral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*cos(c) ^2 + exp_integral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*sin(c))*a/(c^2* cos(c)^2 + c^2*sin(c)^2 + (d*x + c)^2*(cos(c)^2 + sin(c)^2) - 2*(c*cos(c)^ 2 + c*sin(c)^2)*(d*x + c)) - (2*((b*cos(c)^2 + b*sin(c)^2)*(d*x + c)^3 - 3 *(b*c*cos(c)^2 + b*c*sin(c)^2)*(d*x + c)^2 + 3*(b*c^2*cos(c)^2 + b*c^2*sin (c)^2)*(d*x + c))*cos(d*x + c)^3 - 3*(b*c^3*(exp_integral_e(4, I*d*x) + ex p_integral_e(4, -I*d*x))*cos(c)^3 + b*c^3*(exp_integral_e(4, I*d*x) + exp_ integral_e(4, -I*d*x))*cos(c)*sin(c)^2 + b*c^3*(-I*exp_integral_e(4, I*d*x ) + I*exp_integral_e(4, -I*d*x))*sin(c)^3 + b*c^3*(exp_integral_e(4, I*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 564, normalized size of antiderivative = 8.06 \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\frac {a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x^{2} \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) + a d^{3} x^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) - 2 \, a d^{3} x^{2} \operatorname {Si}\left (d x\right ) - 4 \, b x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} + 8 \, a d^{2} x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, a d^{2} x \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, b x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} + 16 \, b x^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, b x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{2} x - 4 \, b x^{2} - 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) - 4 \, a d \tan \left (\frac {1}{2} \, c\right )}{4 \, {\left (d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} + d x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d x^{2}\right )}} \]
1/4*(a*d^3*x^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^3*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a* d^3*x^2*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d^3*x^2*real_p art(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^3*x^2*real_part(c os_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - a*d^3*x^2*imag_part(cos_int egral(d*x))*tan(1/2*d*x)^2 + a*d^3*x^2*imag_part(cos_integral(-d*x))*tan(1 /2*d*x)^2 - 2*a*d^3*x^2*sin_integral(d*x)*tan(1/2*d*x)^2 + a*d^3*x^2*imag_ part(cos_integral(d*x))*tan(1/2*c)^2 - a*d^3*x^2*imag_part(cos_integral(-d *x))*tan(1/2*c)^2 + 2*a*d^3*x^2*sin_integral(d*x)*tan(1/2*c)^2 - 2*a*d^3*x ^2*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d^3*x^2*real_part(cos_int egral(-d*x))*tan(1/2*c) - 2*a*d^2*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^3*x^ 2*imag_part(cos_integral(d*x)) + a*d^3*x^2*imag_part(cos_integral(-d*x)) - 2*a*d^3*x^2*sin_integral(d*x) - 4*b*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a *d^2*x*tan(1/2*d*x)^2 + 8*a*d^2*x*tan(1/2*d*x)*tan(1/2*c) + 2*a*d^2*x*tan( 1/2*c)^2 + 4*b*x^2*tan(1/2*d*x)^2 + 16*b*x^2*tan(1/2*d*x)*tan(1/2*c) + 4*a *d*tan(1/2*d*x)^2*tan(1/2*c) + 4*b*x^2*tan(1/2*c)^2 + 4*a*d*tan(1/2*d*x)*t an(1/2*c)^2 - 2*a*d^2*x - 4*b*x^2 - 4*a*d*tan(1/2*d*x) - 4*a*d*tan(1/2*c)) /(d*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + d*x^2*tan(1/2*d*x)^2 + d*x^2*tan(1/2 *c)^2 + d*x^2)
Timed out. \[ \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^3} \, dx=\int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^3} \,d x \]