\(\int x^3 (a+b x^3) \sin (c+d x) \, dx\) [79]

Optimal result

Integrand size = 17, antiderivative size = 156 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {720 b \cos (c+d x)}{d^7}+\frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {720 b x \sin (c+d x)}{d^6}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2} \]

[Out]

720*b*cos(d*x+c)/d^7+6*a*x*cos(d*x+c)/d^3-360*b*x^2*cos(d*x+c)/d^5-a*x^3*cos(d*x+c)/d+30*b*x^4*cos(d*x+c)/d^3-
b*x^6*cos(d*x+c)/d-6*a*sin(d*x+c)/d^4+720*b*x*sin(d*x+c)/d^6+3*a*x^2*sin(d*x+c)/d^2-120*b*x^3*sin(d*x+c)/d^4+6
*b*x^5*sin(d*x+c)/d^2

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3420, 3377, 2717, 2718} \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=-\frac {6 a \sin (c+d x)}{d^4}+\frac {6 a x \cos (c+d x)}{d^3}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {a x^3 \cos (c+d x)}{d}+\frac {720 b \cos (c+d x)}{d^7}+\frac {720 b x \sin (c+d x)}{d^6}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {30 b x^4 \cos (c+d x)}{d^3}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {b x^6 \cos (c+d x)}{d} \]

[In]

Int[x^3*(a + b*x^3)*Sin[c + d*x],x]

[Out]

(720*b*Cos[c + d*x])/d^7 + (6*a*x*Cos[c + d*x])/d^3 - (360*b*x^2*Cos[c + d*x])/d^5 - (a*x^3*Cos[c + d*x])/d +
(30*b*x^4*Cos[c + d*x])/d^3 - (b*x^6*Cos[c + d*x])/d - (6*a*Sin[c + d*x])/d^4 + (720*b*x*Sin[c + d*x])/d^6 + (
3*a*x^2*Sin[c + d*x])/d^2 - (120*b*x^3*Sin[c + d*x])/d^4 + (6*b*x^5*Sin[c + d*x])/d^2

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3420

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegran
d[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]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^3 \sin (c+d x)+b x^6 \sin (c+d x)\right ) \, dx \\ & = a \int x^3 \sin (c+d x) \, dx+b \int x^6 \sin (c+d x) \, dx \\ & = -\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^6 \cos (c+d x)}{d}+\frac {(3 a) \int x^2 \cos (c+d x) \, dx}{d}+\frac {(6 b) \int x^5 \cos (c+d x) \, dx}{d} \\ & = -\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^6 \cos (c+d x)}{d}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {(6 a) \int x \sin (c+d x) \, dx}{d^2}-\frac {(30 b) \int x^4 \sin (c+d x) \, dx}{d^2} \\ & = \frac {6 a x \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {(6 a) \int \cos (c+d x) \, dx}{d^3}-\frac {(120 b) \int x^3 \cos (c+d x) \, dx}{d^3} \\ & = \frac {6 a x \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}+\frac {(360 b) \int x^2 \sin (c+d x) \, dx}{d^4} \\ & = \frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}+\frac {(720 b) \int x \cos (c+d x) \, dx}{d^5} \\ & = \frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {720 b x \sin (c+d x)}{d^6}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {(720 b) \int \sin (c+d x) \, dx}{d^6} \\ & = \frac {720 b \cos (c+d x)}{d^7}+\frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {720 b x \sin (c+d x)}{d^6}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.65 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {-\left (\left (a d^4 x \left (-6+d^2 x^2\right )+b \left (-720+360 d^2 x^2-30 d^4 x^4+d^6 x^6\right )\right ) \cos (c+d x)\right )+3 d \left (a d^2 \left (-2+d^2 x^2\right )+2 b x \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^7} \]

[In]

Integrate[x^3*(a + b*x^3)*Sin[c + d*x],x]

[Out]

(-((a*d^4*x*(-6 + d^2*x^2) + b*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + 3*d*(a*d^2*(-2 + d
^2*x^2) + 2*b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^7

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {\left (b \,x^{6} d^{6}+a \,d^{6} x^{3}-30 b \,x^{4} d^{4}-6 a \,d^{4} x +360 d^{2} x^{2} b -720 b \right ) \cos \left (d x +c \right )}{d^{7}}+\frac {3 \left (2 b \,d^{4} x^{5}+a \,d^{4} x^{2}-40 b \,d^{2} x^{3}-2 a \,d^{2}+240 b x \right ) \sin \left (d x +c \right )}{d^{6}}\)	\(106\)
parallelrisch	\(\frac {\left (x^{3} \left (b \,x^{3}+a \right ) d^{6}-6 x \left (5 b \,x^{3}+a \right ) d^{4}+360 d^{2} x^{2} b -1440 b \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (x^{2} \left (2 b \,x^{3}+a \right ) d^{4}+\left (-40 b \,x^{3}-2 a \right ) d^{2}+240 b x \right ) d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (x^{2} \left (b \,x^{3}+a \right ) d^{4}+\left (-30 b \,x^{3}-6 a \right ) d^{2}+360 b x \right ) x \,d^{2}}{d^{7} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(159\)
norman	\(\frac {\frac {1440 b}{d^{7}}+\frac {a \,x^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \,x^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {12 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}+\frac {6 a x}{d^{3}}-\frac {a \,x^{3}}{d}-\frac {360 b \,x^{2}}{d^{5}}+\frac {30 b \,x^{4}}{d^{3}}-\frac {b \,x^{6}}{d}-\frac {6 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{3}}+\frac {6 a \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}+\frac {1440 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{6}}+\frac {360 b \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{5}}-\frac {240 b \,x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}-\frac {30 b \,x^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{3}}+\frac {12 b \,x^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\)	\(248\)
meijerg	\(\frac {64 b \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {x \left (d^{2}\right )^{\frac {7}{2}} \left (\frac {21}{8} d^{4} x^{4}-\frac {105}{2} d^{2} x^{2}+315\right ) \cos \left (d x \right )}{28 \sqrt {\pi }\, d^{6}}-\frac {\left (d^{2}\right )^{\frac {7}{2}} \left (-\frac {7}{16} d^{6} x^{6}+\frac {105}{8} d^{4} x^{4}-\frac {315}{2} d^{2} x^{2}+315\right ) \sin \left (d x \right )}{28 \sqrt {\pi }\, d^{7}}\right )}{d^{6} \sqrt {d^{2}}}+\frac {64 b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {45}{4 \sqrt {\pi }}+\frac {\left (-\frac {1}{16} d^{6} x^{6}+\frac {15}{8} d^{4} x^{4}-\frac {45}{2} d^{2} x^{2}+45\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}+\frac {x d \left (\frac {3}{8} d^{4} x^{4}-\frac {15}{2} d^{2} x^{2}+45\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{7}}+\frac {8 a \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 d^{2} x^{2}}{2}+3\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}-\frac {d x \left (-\frac {d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}+\frac {8 a \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {x d \left (-\frac {5 d^{2} x^{2}}{2}+15\right ) \cos \left (d x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 d^{2} x^{2}}{2}+15\right ) \sin \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}\)	\(288\)
parts	\(-\frac {b \,x^{6} \cos \left (d x +c \right )}{d}-\frac {a \,x^{3} \cos \left (d x +c \right )}{d}+\frac {\frac {3 a \,c^{2} \sin \left (d x +c \right )}{d^{2}}-\frac {6 a c \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+\frac {3 a \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}-\frac {6 b \,c^{5} \sin \left (d x +c \right )}{d^{5}}+\frac {30 b \,c^{4} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{5}}-\frac {60 b \,c^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{5}}+\frac {60 b \,c^{2} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{5}}-\frac {30 b c \left (\left (d x +c \right )^{4} \sin \left (d x +c \right )+4 \left (d x +c \right )^{3} \cos \left (d x +c \right )-12 \left (d x +c \right )^{2} \sin \left (d x +c \right )+24 \sin \left (d x +c \right )-24 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{5}}+\frac {6 b \left (\left (d x +c \right )^{5} \sin \left (d x +c \right )+5 \left (d x +c \right )^{4} \cos \left (d x +c \right )-20 \left (d x +c \right )^{3} \sin \left (d x +c \right )-60 \left (d x +c \right )^{2} \cos \left (d x +c \right )+120 \cos \left (d x +c \right )+120 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{5}}}{d^{2}}\)	\(427\)
derivativedivides	\(\frac {a \,c^{3} \cos \left (d x +c \right )+3 a \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )-3 a c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )-\frac {b \,c^{6} \cos \left (d x +c \right )}{d^{3}}-\frac {6 b \,c^{5} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {15 b \,c^{4} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {20 b \,c^{3} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {15 b \,c^{2} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}}{d^{4}}\)	\(556\)
default	\(\frac {a \,c^{3} \cos \left (d x +c \right )+3 a \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )-3 a c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )-\frac {b \,c^{6} \cos \left (d x +c \right )}{d^{3}}-\frac {6 b \,c^{5} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {15 b \,c^{4} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {20 b \,c^{3} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {15 b \,c^{2} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}}{d^{4}}\)	\(556\)

[In]

int(x^3*(b*x^3+a)*sin(d*x+c),x,method=_RETURNVERBOSE)

[Out]

-(b*d^6*x^6+a*d^6*x^3-30*b*d^4*x^4-6*a*d^4*x+360*b*d^2*x^2-720*b)/d^7*cos(d*x+c)+3/d^6*(2*b*d^4*x^5+a*d^4*x^2-
40*b*d^2*x^3-2*a*d^2+240*b*x)*sin(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.67 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=-\frac {{\left (b d^{6} x^{6} + a d^{6} x^{3} - 30 \, b d^{4} x^{4} - 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 720 \, b\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} - 40 \, b d^{3} x^{3} - 2 \, a d^{3} + 240 \, b d x\right )} \sin \left (d x + c\right )}{d^{7}} \]

[In]

integrate(x^3*(b*x^3+a)*sin(d*x+c),x, algorithm="fricas")

[Out]

-((b*d^6*x^6 + a*d^6*x^3 - 30*b*d^4*x^4 - 6*a*d^4*x + 360*b*d^2*x^2 - 720*b)*cos(d*x + c) - 3*(2*b*d^5*x^5 + a
*d^5*x^2 - 40*b*d^3*x^3 - 2*a*d^3 + 240*b*d*x)*sin(d*x + c))/d^7

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.19 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=\begin {cases} - \frac {a x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 a x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 a \sin {\left (c + d x \right )}}{d^{4}} - \frac {b x^{6} \cos {\left (c + d x \right )}}{d} + \frac {6 b x^{5} \sin {\left (c + d x \right )}}{d^{2}} + \frac {30 b x^{4} \cos {\left (c + d x \right )}}{d^{3}} - \frac {120 b x^{3} \sin {\left (c + d x \right )}}{d^{4}} - \frac {360 b x^{2} \cos {\left (c + d x \right )}}{d^{5}} + \frac {720 b x \sin {\left (c + d x \right )}}{d^{6}} + \frac {720 b \cos {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{7}}{7}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*(b*x**3+a)*sin(d*x+c),x)

[Out]

Piecewise((-a*x**3*cos(c + d*x)/d + 3*a*x**2*sin(c + d*x)/d**2 + 6*a*x*cos(c + d*x)/d**3 - 6*a*sin(c + d*x)/d*
*4 - b*x**6*cos(c + d*x)/d + 6*b*x**5*sin(c + d*x)/d**2 + 30*b*x**4*cos(c + d*x)/d**3 - 120*b*x**3*sin(c + d*x
)/d**4 - 360*b*x**2*cos(c + d*x)/d**5 + 720*b*x*sin(c + d*x)/d**6 + 720*b*cos(c + d*x)/d**7, Ne(d, 0)), ((a*x*
*4/4 + b*x**7/7)*sin(c), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (156) = 312\).

Time = 0.22 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.88 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {a c^{3} \cos \left (d x + c\right ) - \frac {b c^{6} \cos \left (d x + c\right )}{d^{3}} - 3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c^{2} + \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{5}}{d^{3}} + 3 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a c - \frac {15 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c^{4}}{d^{3}} - {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a + \frac {20 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b c^{3}}{d^{3}} - \frac {15 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d^{3}} + \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \, {\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b c}{d^{3}} - \frac {{\left ({\left ({\left (d x + c\right )}^{6} - 30 \, {\left (d x + c\right )}^{4} + 360 \, {\left (d x + c\right )}^{2} - 720\right )} \cos \left (d x + c\right ) - 6 \, {\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \sin \left (d x + c\right )\right )} b}{d^{3}}}{d^{4}} \]

[In]

integrate(x^3*(b*x^3+a)*sin(d*x+c),x, algorithm="maxima")

[Out]

(a*c^3*cos(d*x + c) - b*c^6*cos(d*x + c)/d^3 - 3*((d*x + c)*cos(d*x + c) - sin(d*x + c))*a*c^2 + 6*((d*x + c)*
cos(d*x + c) - sin(d*x + c))*b*c^5/d^3 + 3*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*c - 1
5*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b*c^4/d^3 - (((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x
+ c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*a + 20*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 -
2)*sin(d*x + c))*b*c^3/d^3 - 15*(((d*x + c)^4 - 12*(d*x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6
*c)*sin(d*x + c))*b*c^2/d^3 + 6*(((d*x + c)^5 - 20*(d*x + c)^3 + 120*d*x + 120*c)*cos(d*x + c) - 5*((d*x + c)^
4 - 12*(d*x + c)^2 + 24)*sin(d*x + c))*b*c/d^3 - (((d*x + c)^6 - 30*(d*x + c)^4 + 360*(d*x + c)^2 - 720)*cos(d
*x + c) - 6*((d*x + c)^5 - 20*(d*x + c)^3 + 120*d*x + 120*c)*sin(d*x + c))*b/d^3)/d^4

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.68 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=-\frac {{\left (b d^{6} x^{6} + a d^{6} x^{3} - 30 \, b d^{4} x^{4} - 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 720 \, b\right )} \cos \left (d x + c\right )}{d^{7}} + \frac {3 \, {\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} - 40 \, b d^{3} x^{3} - 2 \, a d^{3} + 240 \, b d x\right )} \sin \left (d x + c\right )}{d^{7}} \]

[In]

integrate(x^3*(b*x^3+a)*sin(d*x+c),x, algorithm="giac")

[Out]

-(b*d^6*x^6 + a*d^6*x^3 - 30*b*d^4*x^4 - 6*a*d^4*x + 360*b*d^2*x^2 - 720*b)*cos(d*x + c)/d^7 + 3*(2*b*d^5*x^5
+ a*d^5*x^2 - 40*b*d^3*x^3 - 2*a*d^3 + 240*b*d*x)*sin(d*x + c)/d^7

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.97 \[ \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {d^4\,\left (6\,a\,x\,\cos \left (c+d\,x\right )+30\,b\,x^4\,\cos \left (c+d\,x\right )\right )+720\,b\,\cos \left (c+d\,x\right )-d^6\,\left (a\,x^3\,\cos \left (c+d\,x\right )+b\,x^6\,\cos \left (c+d\,x\right )\right )+d^5\,\left (3\,a\,x^2\,\sin \left (c+d\,x\right )+6\,b\,x^5\,\sin \left (c+d\,x\right )\right )-d^3\,\left (6\,a\,\sin \left (c+d\,x\right )+120\,b\,x^3\,\sin \left (c+d\,x\right )\right )+720\,b\,d\,x\,\sin \left (c+d\,x\right )-360\,b\,d^2\,x^2\,\cos \left (c+d\,x\right )}{d^7} \]

[In]

int(x^3*sin(c + d*x)*(a + b*x^3),x)

[Out]

(d^4*(6*a*x*cos(c + d*x) + 30*b*x^4*cos(c + d*x)) + 720*b*cos(c + d*x) - d^6*(a*x^3*cos(c + d*x) + b*x^6*cos(c
+ d*x)) + d^5*(3*a*x^2*sin(c + d*x) + 6*b*x^5*sin(c + d*x)) - d^3*(6*a*sin(c + d*x) + 120*b*x^3*sin(c + d*x))
+ 720*b*d*x*sin(c + d*x) - 360*b*d^2*x^2*cos(c + d*x))/d^7