Integrand size = 20, antiderivative size = 233 \[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=-\frac {a f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b d f \cos \left (c+\frac {d}{x}\right )}{2 e^3 \left (f+\frac {e}{x}\right )}-\frac {b d \cos \left (c-\frac {d f}{e}\right ) \operatorname {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b d^2 f \operatorname {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \sin \left (c-\frac {d f}{e}\right )}{2 e^4}-\frac {b f \sin \left (c+\frac {d}{x}\right )}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{2 e^4}+\frac {b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3} \]
-1/2*a*f/e^2/(f+e/x)^2+a/e^2/(f+e/x)-b*d*Ci(d*(f/e+1/x))*cos(c-d*f/e)/e^3- 1/2*b*d*f*cos(c+d/x)/e^3/(f+e/x)-1/2*b*d^2*f*cos(c-d*f/e)*Si(d*(f/e+1/x))/ e^4-1/2*b*d^2*f*Ci(d*(f/e+1/x))*sin(c-d*f/e)/e^4+b*d*Si(d*(f/e+1/x))*sin(c -d*f/e)/e^3-1/2*b*f*sin(c+d/x)/e^2/(f+e/x)^2+b*sin(c+d/x)/e^2/(f+e/x)
Time = 1.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=-\frac {b d \operatorname {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \left (2 e \cos \left (c-\frac {d f}{e}\right )+d f \sin \left (c-\frac {d f}{e}\right )\right )+\frac {e \left (a e^3+b d f^2 x (e+f x) \cos \left (c+\frac {d}{x}\right )-b e f x (2 e+f x) \sin \left (c+\frac {d}{x}\right )\right )}{f (e+f x)^2}+b d \left (d f \cos \left (c-\frac {d f}{e}\right )-2 e \sin \left (c-\frac {d f}{e}\right )\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{2 e^4} \]
-1/2*(b*d*CosIntegral[d*(f/e + x^(-1))]*(2*e*Cos[c - (d*f)/e] + d*f*Sin[c - (d*f)/e]) + (e*(a*e^3 + b*d*f^2*x*(e + f*x)*Cos[c + d/x] - b*e*f*x*(2*e + f*x)*Sin[c + d/x]))/(f*(e + f*x)^2) + b*d*(d*f*Cos[c - (d*f)/e] - 2*e*Si n[c - (d*f)/e])*SinIntegral[d*(f/e + x^(-1))])/e^4
Time = 0.66 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3912, 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.
| \(\displaystyle \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle -\int \left (\frac {a+b \sin \left (c+\frac {d}{x}\right )}{e \left (\frac {e}{x}+f\right )^2}-\frac {f \left (a+b \sin \left (c+\frac {d}{x}\right )\right )}{e \left (\frac {e}{x}+f\right )^3}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a f}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {b d^2 f \sin \left (c-\frac {d f}{e}\right ) \operatorname {CosIntegral}\left (\frac {f d}{e}+\frac {d}{x}\right )}{2 e^4}-\frac {b d \cos \left (c-\frac {d f}{e}\right ) \operatorname {CosIntegral}\left (\frac {f d}{e}+\frac {d}{x}\right )}{e^3}-\frac {b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {f d}{e}+\frac {d}{x}\right )}{2 e^4}+\frac {b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {f d}{e}+\frac {d}{x}\right )}{e^3}-\frac {b d f \cos \left (c+\frac {d}{x}\right )}{2 e^3 \left (\frac {e}{x}+f\right )}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {b f \sin \left (c+\frac {d}{x}\right )}{2 e^2 \left (\frac {e}{x}+f\right )^2}\) |
-1/2*(a*f)/(e^2*(f + e/x)^2) + a/(e^2*(f + e/x)) - (b*d*f*Cos[c + d/x])/(2 *e^3*(f + e/x)) - (b*d*Cos[c - (d*f)/e]*CosIntegral[(d*f)/e + d/x])/e^3 - (b*d^2*f*CosIntegral[(d*f)/e + d/x]*Sin[c - (d*f)/e])/(2*e^4) - (b*f*Sin[c + d/x])/(2*e^2*(f + e/x)^2) + (b*Sin[c + d/x])/(e^2*(f + e/x)) - (b*d^2*f *Cos[c - (d*f)/e]*SinIntegral[(d*f)/e + d/x])/(2*e^4) + (b*d*Sin[c - (d*f) /e]*SinIntegral[(d*f)/e + d/x])/e^3
3.3.93.3.1 Defintions of rubi rules used
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p , 0] && IntegerQ[1/n]
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.82
| method | result | size |
| risch | \(-\frac {a}{2 f \left (f x +e \right )^{2}}+\frac {i b \,d^{2} {\mathrm e}^{-\frac {i \left (c e -d f \right )}{e}} \operatorname {Ei}_{1}\left (\frac {i d}{x}+i c -\frac {i \left (c e -d f \right )}{e}\right ) f}{4 e^{4}}+\frac {b d \,{\mathrm e}^{-\frac {i \left (c e -d f \right )}{e}} \operatorname {Ei}_{1}\left (\frac {i d}{x}+i c -\frac {i \left (c e -d f \right )}{e}\right )}{2 e^{3}}-\frac {i b \,d^{2} {\mathrm e}^{\frac {i \left (c e -d f \right )}{e}} \operatorname {Ei}_{1}\left (-\frac {i d}{x}-i c -\frac {-i c e +i f d}{e}\right ) f}{4 e^{4}}+\frac {b d \,{\mathrm e}^{\frac {i \left (c e -d f \right )}{e}} \operatorname {Ei}_{1}\left (-\frac {i d}{x}-i c -\frac {-i c e +i f d}{e}\right )}{2 e^{3}}+\frac {i b x \left (2 i d^{3} f^{4} x^{3}+6 i d^{3} e \,f^{3} x^{2}+6 i d^{3} e^{2} f^{2} x +2 i d^{3} e^{3} f \right ) \cos \left (\frac {c x +d}{x}\right )}{4 e^{3} \left (f x +e \right )^{2} \left (d^{2} x^{2} f^{2}+2 f e x \,d^{2}+d^{2} e^{2}\right )}-\frac {b x \left (-2 d^{2} f^{3} x^{3}-8 d^{2} e \,f^{2} x^{2}-10 d^{2} e^{2} f x -4 d^{2} e^{3}\right ) \sin \left (\frac {c x +d}{x}\right )}{4 e^{2} \left (f x +e \right )^{2} \left (d^{2} x^{2} f^{2}+2 f e x \,d^{2}+d^{2} e^{2}\right )}\) | \(423\) |
| parts | \(-\frac {a}{2 f \left (f x +e \right )^{2}}-b d \left (\frac {-\frac {\sin \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}+\frac {\frac {\operatorname {Si}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}+\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{e}+\frac {\left (c e -d f \right ) \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}-\frac {\frac {\operatorname {Si}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )}{e}-c \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}-\frac {\frac {\operatorname {Si}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )\right )\) | \(455\) |
| derivativedivides | \(-d \left (-\frac {a}{e^{2} \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )}-\frac {\left (c e -d f \right ) a}{2 e^{2} \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2}}+\frac {b \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}+\frac {-\frac {\operatorname {Si}\left (-\frac {d}{x}-c -\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}+\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}}{e}\right )}{e}+\frac {b \left (c e -d f \right ) \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}-\frac {-\frac {\operatorname {Si}\left (-\frac {d}{x}-c -\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )}{e}+\frac {c a}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}-b c \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}-\frac {-\frac {\operatorname {Si}\left (-\frac {d}{x}-c -\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )\right )\) | \(542\) |
| default | \(-d \left (-\frac {a}{e^{2} \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )}-\frac {\left (c e -d f \right ) a}{2 e^{2} \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2}}+\frac {b \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}+\frac {-\frac {\operatorname {Si}\left (-\frac {d}{x}-c -\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}+\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}}{e}\right )}{e}+\frac {b \left (c e -d f \right ) \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}-\frac {-\frac {\operatorname {Si}\left (-\frac {d}{x}-c -\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )}{e}+\frac {c a}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}-b c \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (-c e +d f +e \left (c +\frac {d}{x}\right )\right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (-c e +d f +e \left (c +\frac {d}{x}\right )\right ) e}-\frac {-\frac {\operatorname {Si}\left (-\frac {d}{x}-c -\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\operatorname {Ci}\left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )\right )\) | \(542\) |
-1/2*a/f/(f*x+e)^2+1/4*I*b*d^2/e^4*exp(-I*(c*e-d*f)/e)*Ei(1,I*d/x+I*c-I*(c *e-d*f)/e)*f+1/2*b*d/e^3*exp(-I*(c*e-d*f)/e)*Ei(1,I*d/x+I*c-I*(c*e-d*f)/e) -1/4*I*b*d^2*exp(I*(c*e-d*f)/e)*Ei(1,-I*d/x-I*c-(-I*c*e+I*f*d)/e)/e^4*f+1/ 2*b*d*exp(I*(c*e-d*f)/e)*Ei(1,-I*d/x-I*c-(-I*c*e+I*f*d)/e)/e^3+1/4*I*b/e^3 *x*(2*I*d^3*f^4*x^3+2*I*d^3*e^3*f+6*I*d^3*e*f^3*x^2+6*I*d^3*e^2*f^2*x)/(f* x+e)^2/(d^2*f^2*x^2+2*d^2*e*f*x+d^2*e^2)*cos((c*x+d)/x)-1/4*b/e^2*x*(-2*d^ 2*f^3*x^3-8*d^2*e*f^2*x^2-10*d^2*e^2*f*x-4*d^2*e^3)/(f*x+e)^2/(d^2*f^2*x^2 +2*d^2*e*f*x+d^2*e^2)*sin((c*x+d)/x)
Time = 0.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=-\frac {a e^{4} + {\left (2 \, {\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) + {\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname {Si}\left (\frac {d f x + d e}{e x}\right )\right )} \cos \left (-\frac {c e - d f}{e}\right ) + {\left (b d e f^{3} x^{2} + b d e^{2} f^{2} x\right )} \cos \left (\frac {c x + d}{x}\right ) - {\left ({\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) - 2 \, {\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname {Si}\left (\frac {d f x + d e}{e x}\right )\right )} \sin \left (-\frac {c e - d f}{e}\right ) - {\left (b e^{2} f^{2} x^{2} + 2 \, b e^{3} f x\right )} \sin \left (\frac {c x + d}{x}\right )}{2 \, {\left (e^{4} f^{3} x^{2} + 2 \, e^{5} f^{2} x + e^{6} f\right )}} \]
-1/2*(a*e^4 + (2*(b*d*e*f^3*x^2 + 2*b*d*e^2*f^2*x + b*d*e^3*f)*cos_integra l((d*f*x + d*e)/(e*x)) + (b*d^2*f^4*x^2 + 2*b*d^2*e*f^3*x + b*d^2*e^2*f^2) *sin_integral((d*f*x + d*e)/(e*x)))*cos(-(c*e - d*f)/e) + (b*d*e*f^3*x^2 + b*d*e^2*f^2*x)*cos((c*x + d)/x) - ((b*d^2*f^4*x^2 + 2*b*d^2*e*f^3*x + b*d ^2*e^2*f^2)*cos_integral((d*f*x + d*e)/(e*x)) - 2*(b*d*e*f^3*x^2 + 2*b*d*e ^2*f^2*x + b*d*e^3*f)*sin_integral((d*f*x + d*e)/(e*x)))*sin(-(c*e - d*f)/ e) - (b*e^2*f^2*x^2 + 2*b*e^3*f*x)*sin((c*x + d)/x))/(e^4*f^3*x^2 + 2*e^5* f^2*x + e^6*f)
\[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=\int \frac {a + b \sin {\left (c + \frac {d}{x} \right )}}{\left (e + f x\right )^{3}}\, dx \]
\[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=\int { \frac {b \sin \left (c + \frac {d}{x}\right ) + a}{{\left (f x + e\right )}^{3}} \,d x } \]
b*(integrate(1/2*sin((c*x + d)/x)/(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3 ), x) + integrate(1/2*sin((c*x + d)/x)/((f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3)*cos((c*x + d)/x)^2 + (f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3)*sin ((c*x + d)/x)^2), x)) - 1/2*a/(f^3*x^2 + 2*e*f^2*x + e^2*f)
Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (223) = 446\).
Time = 0.38 (sec) , antiderivative size = 1501, normalized size of antiderivative = 6.44 \[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=\text {Too large to display} \]
-1/2*(b*c^2*d^3*e^2*f*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e)*sin((c* e - d*f)/e) - 2*b*c*d^4*e*f^2*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e) *sin((c*e - d*f)/e) + b*d^5*f^3*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/ e)*sin((c*e - d*f)/e) - b*c^2*d^3*e^2*f*cos((c*e - d*f)/e)*sin_integral((c *e - d*f - (c*x + d)*e/x)/e) + 2*b*c*d^4*e*f^2*cos((c*e - d*f)/e)*sin_inte gral((c*e - d*f - (c*x + d)*e/x)/e) - b*d^5*f^3*cos((c*e - d*f)/e)*sin_int egral((c*e - d*f - (c*x + d)*e/x)/e) + 2*b*c^2*d^2*e^3*cos((c*e - d*f)/e)* cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e) - 4*b*c*d^3*e^2*f*cos((c*e - d*f)/e)*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e) + 2*b*d^4*e*f^2*cos(( c*e - d*f)/e)*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e) - 2*(c*x + d)*b *c*d^3*e^2*f*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e)*sin((c*e - d*f)/ e)/x + 2*(c*x + d)*b*d^4*e*f^2*cos_integral(-(c*e - d*f - (c*x + d)*e/x)/e )*sin((c*e - d*f)/e)/x + 2*(c*x + d)*b*c*d^3*e^2*f*cos((c*e - d*f)/e)*sin_ integral((c*e - d*f - (c*x + d)*e/x)/e)/x - 2*(c*x + d)*b*d^4*e*f^2*cos((c *e - d*f)/e)*sin_integral((c*e - d*f - (c*x + d)*e/x)/e)/x + 2*b*c^2*d^2*e ^3*sin((c*e - d*f)/e)*sin_integral((c*e - d*f - (c*x + d)*e/x)/e) - 4*b*c* d^3*e^2*f*sin((c*e - d*f)/e)*sin_integral((c*e - d*f - (c*x + d)*e/x)/e) + 2*b*d^4*e*f^2*sin((c*e - d*f)/e)*sin_integral((c*e - d*f - (c*x + d)*e/x) /e) - b*c*d^3*e^2*f*cos((c*x + d)/x) + b*d^4*e*f^2*cos((c*x + d)/x) - 4*(c *x + d)*b*c*d^2*e^3*cos((c*e - d*f)/e)*cos_integral(-(c*e - d*f - (c*x ...
Timed out. \[ \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^3} \, dx=\int \frac {a+b\,\sin \left (c+\frac {d}{x}\right )}{{\left (e+f\,x\right )}^3} \,d x \]