Integrand size = 16, antiderivative size = 194 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 (b c-a d) \cos \left (\frac {b}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \cos \left (\frac {3 b}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {3 (b c-a d) \sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2} \]
3/4*(-a*d+b*c)*Ci((-a*d+b*c)/d/(d*x+c))*cos(b/d)/d^2-3/4*(-a*d+b*c)*Ci(3*( -a*d+b*c)/d/(d*x+c))*cos(3*b/d)/d^2+3/4*(-a*d+b*c)*Si((-a*d+b*c)/d/(d*x+c) )*sin(b/d)/d^2-3/4*(-a*d+b*c)*Si(3*(-a*d+b*c)/d/(d*x+c))*sin(3*b/d)/d^2+(d *x+c)*sin((b*x+a)/(d*x+c))^3/d
Result contains complex when optimal does not.
Time = 7.78 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.57 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 i c d e^{-\frac {i (a+b x)}{c+d x}}-3 i c d e^{\frac {i (a+b x)}{c+d x}}-i c d e^{-\frac {3 i (a+b x)}{c+d x}}+i c d e^{\frac {3 i (a+b x)}{c+d x}}+6 d^2 x \cos \left (\frac {-b c+a d}{d (c+d x)}\right ) \sin \left (\frac {b}{d}\right )-2 d^2 x \cos \left (\frac {3 (-b c+a d)}{d (c+d x)}\right ) \sin \left (\frac {3 b}{d}\right )+6 d^2 x \cos \left (\frac {b}{d}\right ) \sin \left (\frac {-b c+a d}{d (c+d x)}\right )-2 d^2 x \cos \left (\frac {3 b}{d}\right ) \sin \left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+3 (b c-a d) \left (-\cos \left (\frac {3 b}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )+\cos \left (\frac {b}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c-a d}{c d+d^2 x}\right )+\operatorname {CosIntegral}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cos \left (\frac {b}{d}\right )-i \sin \left (\frac {b}{d}\right )\right )+i \operatorname {CosIntegral}\left (\frac {b c-a d}{c d+d^2 x}\right ) \sin \left (\frac {b}{d}\right )-\operatorname {CosIntegral}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right ) \left (\cos \left (\frac {3 b}{d}\right )-i \sin \left (\frac {3 b}{d}\right )\right )-i \operatorname {CosIntegral}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right ) \sin \left (\frac {3 b}{d}\right )+i \cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {-b c+a d}{d (c+d x)}\right )-\sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {-b c+a d}{d (c+d x)}\right )-i \cos \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+\sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )-i \cos \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )-\sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )+i \cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{c d+d^2 x}\right )+\sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{8 d^2} \]
(((3*I)*c*d)/E^((I*(a + b*x))/(c + d*x)) - (3*I)*c*d*E^((I*(a + b*x))/(c + d*x)) - (I*c*d)/E^(((3*I)*(a + b*x))/(c + d*x)) + I*c*d*E^(((3*I)*(a + b* x))/(c + d*x)) + 6*d^2*x*Cos[(-(b*c) + a*d)/(d*(c + d*x))]*Sin[b/d] - 2*d^ 2*x*Cos[(3*(-(b*c) + a*d))/(d*(c + d*x))]*Sin[(3*b)/d] + 6*d^2*x*Cos[b/d]* Sin[(-(b*c) + a*d)/(d*(c + d*x))] - 2*d^2*x*Cos[(3*b)/d]*Sin[(3*(-(b*c) + a*d))/(d*(c + d*x))] + 3*(b*c - a*d)*(-(Cos[(3*b)/d]*CosIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x)]) + Cos[b/d]*CosIntegral[(b*c - a*d)/(c*d + d^2*x)] + CosIntegral[(-(b*c) + a*d)/(d*(c + d*x))]*(Cos[b/d] - I*Sin[b/d]) + I*Cos Integral[(b*c - a*d)/(c*d + d^2*x)]*Sin[b/d] - CosIntegral[(3*(-(b*c) + a* d))/(d*(c + d*x))]*(Cos[(3*b)/d] - I*Sin[(3*b)/d]) - I*CosIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x)]*Sin[(3*b)/d] + I*Cos[b/d]*SinIntegral[(-(b*c) + a* d)/(d*(c + d*x))] - Sin[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*x))] - I *Cos[(3*b)/d]*SinIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] + Sin[(3*b)/d] *SinIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - I*Cos[(3*b)/d]*SinIntegra l[(3*b*c - 3*a*d)/(c*d + d^2*x)] - Sin[(3*b)/d]*SinIntegral[(3*b*c - 3*a*d )/(c*d + d^2*x)] + I*Cos[b/d]*SinIntegral[(b*c - a*d)/(c*d + d^2*x)] + Sin [b/d]*SinIntegral[(b*c - a*d)/(c*d + d^2*x)]))/(8*d^2)
Time = 0.48 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5074, 3042, 3794, 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.
| \(\displaystyle \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 5074 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sin ^3\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sin \left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {-\frac {3 (b c-a d) \int \left (\frac {1}{4} (c+d x) \cos \left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )-\frac {1}{4} (c+d x) \cos \left (\frac {3 b}{d}-\frac {3 (b c-a d)}{d (c+d x)}\right )\right )d\frac {1}{c+d x}}{d}-\left ((c+d x) \sin ^3\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {3 (b c-a d) \left (\frac {1}{4} \cos \left (\frac {b}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )-\frac {1}{4} \cos \left (\frac {3 b}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )+\frac {1}{4} \sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )-\frac {1}{4} \sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )\right )}{d}-\left ((c+d x) \sin ^3\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\) |
-((-((c + d*x)*Sin[b/d - (b*c - a*d)/(d*(c + d*x))]^3) - (3*(b*c - a*d)*(( Cos[b/d]*CosIntegral[(b*c - a*d)/(d*(c + d*x))])/4 - (Cos[(3*b)/d]*CosInte gral[(3*(b*c - a*d))/(d*(c + d*x))])/4 + (Sin[b/d]*SinIntegral[(b*c - a*d) /(d*(c + d*x))])/4 - (Sin[(3*b)/d]*SinIntegral[(3*(b*c - a*d))/(d*(c + d*x ))])/4))/d)/d)
3.1.38.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Int[Sin[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> Simp[-d^(-1) Subst[Int[Sin[b*(e/d) - e*(b*c - a*d)*(x/d)]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a* d, 0]
Time = 1.73 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(-\frac {\left (a d -c b \right ) \left (-\frac {d^{2} \left (-\frac {3 \sin \left (\frac {3 a d -3 c b}{d \left (d x +c \right )}+\frac {3 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {3 b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {3 b}{d}\right )}{d}}{d}\right )}{12}+\frac {3 d^{2} \left (-\frac {\sin \left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {\operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}}{d}\right )}{4}\right )}{d^{2}}\) | \(295\) |
| default | \(-\frac {\left (a d -c b \right ) \left (-\frac {d^{2} \left (-\frac {3 \sin \left (\frac {3 a d -3 c b}{d \left (d x +c \right )}+\frac {3 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {3 b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {3 b}{d}\right )}{d}}{d}\right )}{12}+\frac {3 d^{2} \left (-\frac {\sin \left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {\operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}}{d}\right )}{4}\right )}{d^{2}}\) | \(295\) |
| risch | \(-\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) b c}{4 d^{2}}-\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \pi \,\operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) a}{4 d}+\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \pi \,\operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {3 \,{\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {3 i b}{d}} a}{8 d}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {3 i b}{d}} c b}{8 d^{2}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {i b}{d}} a}{8 d}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {i b}{d}} c b}{8 d^{2}}-\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi b c}{8 d^{2}}-\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) a}{4 d}+\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi a}{8 d}+\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) b c}{4 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {i b}{d}} \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {i b}{d}} \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {3 \sin \left (\frac {x b +a}{d x +c}\right ) x}{4}+\frac {3 \sin \left (\frac {x b +a}{d x +c}\right ) c}{4 d}-\frac {\sin \left (\frac {3 x b +3 a}{d x +c}\right ) x}{4}-\frac {\sin \left (\frac {3 x b +3 a}{d x +c}\right ) c}{4 d}\) | \(668\) |
-1/d^2*(a*d-b*c)*(-1/12*d^2*(-3*sin(3*(a*d-b*c)/d/(d*x+c)+3*b/d)/((b/d+(a* d-b*c)/d/(d*x+c))*d-b)/d+3*(-3*Si(3*(a*d-b*c)/d/(d*x+c))*sin(3*b/d)/d+3*Ci (3*(a*d-b*c)/d/(d*x+c))*cos(3*b/d)/d)/d)+3/4*d^2*(-sin(b/d+(a*d-b*c)/d/(d* x+c))/((b/d+(a*d-b*c)/d/(d*x+c))*d-b)/d+(-Si((a*d-b*c)/d/(d*x+c))*sin(b/d) /d+Ci((a*d-b*c)/d/(d*x+c))*cos(b/d)/d)/d))
Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.09 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 \, {\left (b c - a d\right )} \cos \left (\frac {b}{d}\right ) \operatorname {Ci}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) - 3 \, {\left (b c - a d\right )} \cos \left (\frac {3 \, b}{d}\right ) \operatorname {Ci}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) - 3 \, {\left (b c - a d\right )} \sin \left (\frac {b}{d}\right ) \operatorname {Si}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) + 3 \, {\left (b c - a d\right )} \sin \left (\frac {3 \, b}{d}\right ) \operatorname {Si}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) + 4 \, {\left (d^{2} x - {\left (d^{2} x + c d\right )} \cos \left (\frac {b x + a}{d x + c}\right )^{2} + c d\right )} \sin \left (\frac {b x + a}{d x + c}\right )}{4 \, d^{2}} \]
1/4*(3*(b*c - a*d)*cos(b/d)*cos_integral(-(b*c - a*d)/(d^2*x + c*d)) - 3*( b*c - a*d)*cos(3*b/d)*cos_integral(-3*(b*c - a*d)/(d^2*x + c*d)) - 3*(b*c - a*d)*sin(b/d)*sin_integral(-(b*c - a*d)/(d^2*x + c*d)) + 3*(b*c - a*d)*s in(3*b/d)*sin_integral(-3*(b*c - a*d)/(d^2*x + c*d)) + 4*(d^2*x - (d^2*x + c*d)*cos((b*x + a)/(d*x + c))^2 + c*d)*sin((b*x + a)/(d*x + c)))/d^2
Timed out. \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\text {Timed out} \]
\[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \sin \left (\frac {b x + a}{d x + c}\right )^{3} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1239 vs. \(2 (186) = 372\).
Time = 66.42 (sec) , antiderivative size = 1239, normalized size of antiderivative = 6.39 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\text {Too large to display} \]
1/4*(3*b^3*c^2*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d) - 6*a *b^2*c*d*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d) - 3*(b*x + a)*b^2*c^2*d*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) + 3*a^2*b*d^2*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d) + 6 *(b*x + a)*a*b*c*d^2*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d) /(d*x + c) - 3*(b*x + a)*a^2*d^3*cos(b/d)*cos_integral(-(b - (b*x + a)*d/( d*x + c))/d)/(d*x + c) - 3*b^3*c^2*cos(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/d) + 6*a*b^2*c*d*cos(3*b/d)*cos_integral(-3*(b - (b*x + a) *d/(d*x + c))/d) + 3*(b*x + a)*b^2*c^2*d*cos(3*b/d)*cos_integral(-3*(b - ( b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*a^2*b*d^2*cos(3*b/d)*cos_integral(- 3*(b - (b*x + a)*d/(d*x + c))/d) - 6*(b*x + a)*a*b*c*d^2*cos(3*b/d)*cos_in tegral(-3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) + 3*(b*x + a)*a^2*d^3*c os(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*b^3 *c^2*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d) + 6*a*b^2*c* d*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d) + 3*(b*x + a)*b ^2*c^2*d*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c ) - 3*a^2*b*d^2*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d) - 6*(b*x + a)*a*b*c*d^2*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c ))/d)/(d*x + c) + 3*(b*x + a)*a^2*d^3*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) + 3*b^3*c^2*sin(b/d)*sin_integral((b - (...
Timed out. \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\int {\sin \left (\frac {a+b\,x}{c+d\,x}\right )}^3 \,d x \]