\(\int \frac {(a+b x^3)^2 \sin (c+d x)}{x} \, dx\) [89]

Optimal result

Integrand size = 19, antiderivative size = 161 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\frac {4 a b \cos (c+d x)}{d^3}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {120 b^2 \sin (c+d x)}{d^6}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x) \]

[Out]

4*a*b*cos(d*x+c)/d^3-120*b^2*x*cos(d*x+c)/d^5-2*a*b*x^2*cos(d*x+c)/d+20*b^2*x^3*cos(d*x+c)/d^3-b^2*x^5*cos(d*x
+c)/d+a^2*cos(c)*Si(d*x)+a^2*Ci(d*x)*sin(c)+120*b^2*sin(d*x+c)/d^6+4*a*b*x*sin(d*x+c)/d^2-60*b^2*x^2*sin(d*x+c
)/d^4+5*b^2*x^4*sin(d*x+c)/d^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3420, 3384, 3380, 3383, 3377, 2718, 2717} \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=a^2 \sin (c) \operatorname {CosIntegral}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {4 a b \cos (c+d x)}{d^3}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {b^2 x^5 \cos (c+d x)}{d} \]

[In]

Int[((a + b*x^3)^2*Sin[c + d*x])/x,x]

[Out]

(4*a*b*Cos[c + d*x])/d^3 - (120*b^2*x*Cos[c + d*x])/d^5 - (2*a*b*x^2*Cos[c + d*x])/d + (20*b^2*x^3*Cos[c + d*x
])/d^3 - (b^2*x^5*Cos[c + d*x])/d + a^2*CosIntegral[d*x]*Sin[c] + (120*b^2*Sin[c + d*x])/d^6 + (4*a*b*x*Sin[c
+ d*x])/d^2 - (60*b^2*x^2*Sin[c + d*x])/d^4 + (5*b^2*x^4*Sin[c + d*x])/d^2 + a^2*Cos[c]*SinIntegral[d*x]

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 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 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 (\frac {a^2 \sin (c+d x)}{x}+2 a b x^2 \sin (c+d x)+b^2 x^5 \sin (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\sin (c+d x)}{x} \, dx+(2 a b) \int x^2 \sin (c+d x) \, dx+b^2 \int x^5 \sin (c+d x) \, dx \\ & = -\frac {2 a b x^2 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+\frac {(4 a b) \int x \cos (c+d x) \, dx}{d}+\frac {\left (5 b^2\right ) \int x^4 \cos (c+d x) \, dx}{d}+\left (a^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx+\left (a^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx \\ & = -\frac {2 a b x^2 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)-\frac {(4 a b) \int \sin (c+d x) \, dx}{d^2}-\frac {\left (20 b^2\right ) \int x^3 \sin (c+d x) \, dx}{d^2} \\ & = \frac {4 a b \cos (c+d x)}{d^3}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)-\frac {\left (60 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d^3} \\ & = \frac {4 a b \cos (c+d x)}{d^3}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)+\frac {\left (120 b^2\right ) \int x \sin (c+d x) \, dx}{d^4} \\ & = \frac {4 a b \cos (c+d x)}{d^3}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)+\frac {\left (120 b^2\right ) \int \cos (c+d x) \, dx}{d^5} \\ & = \frac {4 a b \cos (c+d x)}{d^3}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {120 b^2 \sin (c+d x)}{d^6}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=-\frac {b \left (2 a d^2 \left (-2+d^2 x^2\right )+b x \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \cos (c+d x)}{d^5}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {b \left (4 a d^4 x+5 b \left (24-12 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^6}+a^2 \cos (c) \text {Si}(d x) \]

[In]

Integrate[((a + b*x^3)^2*Sin[c + d*x])/x,x]

[Out]

-((b*(2*a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x])/d^5) + a^2*CosIntegral[d*x]*Sin
[c] + (b*(4*a*d^4*x + 5*b*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6 + a^2*Cos[c]*SinIntegral[d*x]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.11

method	result	size
risch	\(-\frac {b^{2} x^{5} \cos \left (d x +c \right )}{d}+\frac {5 b^{2} x^{4} \sin \left (d x +c \right )}{d^{2}}-\frac {i a^{2} {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right )}{2}+\frac {i a^{2} {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2}-\frac {2 a b \,x^{2} \cos \left (d x +c \right )}{d}+\frac {20 b^{2} x^{3} \cos \left (d x +c \right )}{d^{3}}+\frac {4 a b x \sin \left (d x +c \right )}{d^{2}}-\frac {60 b^{2} x^{2} \sin \left (d x +c \right )}{d^{4}}+\frac {4 a b \cos \left (d x +c \right )}{d^{3}}-\frac {120 b^{2} x \cos \left (d x +c \right )}{d^{5}}+\frac {120 b^{2} \sin \left (d x +c \right )}{d^{6}}\)	\(178\)
meijerg	\(\frac {32 b^{2} \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} d^{4} x^{4}-\frac {45}{2} d^{2} x^{2}+45\right ) \cos \left (d x \right )}{12 \sqrt {\pi }}+\frac {x d \left (\frac {3}{8} d^{4} x^{4}-\frac {15}{2} d^{2} x^{2}+45\right ) \sin \left (d x \right )}{12 \sqrt {\pi }}\right )}{d^{6}}+\frac {32 b^{2} \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {x d \left (\frac {7}{8} d^{4} x^{4}-\frac {35}{2} d^{2} x^{2}+105\right ) \cos \left (d x \right )}{28 \sqrt {\pi }}+\frac {\left (\frac {35}{8} d^{4} x^{4}-\frac {105}{2} d^{2} x^{2}+105\right ) \sin \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{6}}+\frac {8 a b \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {x \left (d^{2}\right )^{\frac {3}{2}} \cos \left (d x \right )}{2 \sqrt {\pi }\, d^{2}}-\frac {\left (d^{2}\right )^{\frac {3}{2}} \left (-\frac {3 d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{6 \sqrt {\pi }\, d^{3}}\right )}{d^{2} \sqrt {d^{2}}}+\frac {8 a b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {d^{2} x^{2}}{2}+1\right ) \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {d x \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+\frac {a^{2} \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+a^{2} \cos \left (c \right ) \operatorname {Si}\left (d x \right )\)	\(327\)
derivativedivides	\(a^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )-\frac {6 a b \,c^{2} \cos \left (d x +c \right )}{d^{3}}-\frac {6 a b c \left (c +1\right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {2 \left (c^{2}+c +1\right ) a b \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {6 b^{2} c^{5} \cos \left (d x +c \right )}{d^{6}}+\frac {15 \left (c +1\right ) b^{2} c^{4} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {20 b^{2} c^{3} \left (c^{2}+c +1\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {15 \left (c^{3}+c^{2}+c +1\right ) b^{2} c^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {6 b^{2} c \left (c^{4}+c^{3}+c^{2}+c +1\right ) \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {\left (c^{5}+c^{4}+c^{3}+c^{2}+c +1\right ) b^{2} \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}\)	\(487\)
default	\(a^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )-\frac {6 a b \,c^{2} \cos \left (d x +c \right )}{d^{3}}-\frac {6 a b c \left (c +1\right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {2 \left (c^{2}+c +1\right ) a b \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {6 b^{2} c^{5} \cos \left (d x +c \right )}{d^{6}}+\frac {15 \left (c +1\right ) b^{2} c^{4} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {20 b^{2} c^{3} \left (c^{2}+c +1\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {15 \left (c^{3}+c^{2}+c +1\right ) b^{2} c^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {6 b^{2} c \left (c^{4}+c^{3}+c^{2}+c +1\right ) \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {\left (c^{5}+c^{4}+c^{3}+c^{2}+c +1\right ) b^{2} \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}\)	\(487\)

[In]

int((b*x^3+a)^2*sin(d*x+c)/x,x,method=_RETURNVERBOSE)

[Out]

-b^2*x^5*cos(d*x+c)/d+5*b^2*x^4*sin(d*x+c)/d^2-1/2*I*a^2*exp(-I*c)*Ei(1,I*d*x)+1/2*I*a^2*exp(I*c)*Ei(1,-I*d*x)
-2*a*b*x^2*cos(d*x+c)/d+20*b^2*x^3*cos(d*x+c)/d^3+4*a*b*x*sin(d*x+c)/d^2-60*b^2*x^2*sin(d*x+c)/d^4+4*a*b*cos(d
*x+c)/d^3-120*b^2*x*cos(d*x+c)/d^5+120*b^2*sin(d*x+c)/d^6

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\frac {a^{2} d^{6} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + a^{2} d^{6} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \cos \left (d x + c\right ) + {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x - 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{6}} \]

[In]

integrate((b*x^3+a)^2*sin(d*x+c)/x,x, algorithm="fricas")

[Out]

(a^2*d^6*cos_integral(d*x)*sin(c) + a^2*d^6*cos(c)*sin_integral(d*x) - (b^2*d^5*x^5 + 2*a*b*d^5*x^2 - 20*b^2*d
^3*x^3 - 4*a*b*d^3 + 120*b^2*d*x)*cos(d*x + c) + (5*b^2*d^4*x^4 + 4*a*b*d^4*x - 60*b^2*d^2*x^2 + 120*b^2)*sin(
d*x + c))/d^6

Sympy [A] (verification not implemented)

Time = 3.69 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=a^{2} \sin {\left (c \right )} \operatorname {Ci}{\left (d x \right )} + a^{2} \cos {\left (c \right )} \operatorname {Si}{\left (d x \right )} + 2 a b x^{2} \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 4 a b \left (\begin {cases} \frac {x^{3} \sin {\left (c \right )}}{3} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x \sin {\left (c + d x \right )}}{d} + \frac {\cos {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cos {\left (c \right )}}{2} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) + b^{2} x^{5} \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 5 b^{2} \left (\begin {cases} \frac {x^{6} \sin {\left (c \right )}}{6} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x^{4} \sin {\left (c + d x \right )}}{d} + \frac {4 x^{3} \cos {\left (c + d x \right )}}{d^{2}} - \frac {12 x^{2} \sin {\left (c + d x \right )}}{d^{3}} - \frac {24 x \cos {\left (c + d x \right )}}{d^{4}} + \frac {24 \sin {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\frac {x^{5} \cos {\left (c \right )}}{5} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((b*x**3+a)**2*sin(d*x+c)/x,x)

[Out]

a**2*sin(c)*Ci(d*x) + a**2*cos(c)*Si(d*x) + 2*a*b*x**2*Piecewise((x*sin(c), Eq(d, 0)), (-cos(c + d*x)/d, True)
) - 4*a*b*Piecewise((x**3*sin(c)/3, Eq(d, 0)), (-Piecewise((x*sin(c + d*x)/d + cos(c + d*x)/d**2, Ne(d, 0)), (
x**2*cos(c)/2, True))/d, True)) + b**2*x**5*Piecewise((x*sin(c), Eq(d, 0)), (-cos(c + d*x)/d, True)) - 5*b**2*
Piecewise((x**6*sin(c)/6, Eq(d, 0)), (-Piecewise((x**4*sin(c + d*x)/d + 4*x**3*cos(c + d*x)/d**2 - 12*x**2*sin
(c + d*x)/d**3 - 24*x*cos(c + d*x)/d**4 + 24*sin(c + d*x)/d**5, Ne(d, 0)), (x**5*cos(c)/5, True))/d, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.68 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\frac {{\left (a^{2} {\left (-i \, {\rm Ei}\left (i \, d x\right ) + i \, {\rm Ei}\left (-i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left ({\rm Ei}\left (i \, d x\right ) + {\rm Ei}\left (-i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 2 \, {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x - 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{6}} \]

[In]

integrate((b*x^3+a)^2*sin(d*x+c)/x,x, algorithm="maxima")

[Out]

1/2*((a^2*(-I*Ei(I*d*x) + I*Ei(-I*d*x))*cos(c) + a^2*(Ei(I*d*x) + Ei(-I*d*x))*sin(c))*d^6 - 2*(b^2*d^5*x^5 + 2
*a*b*d^5*x^2 - 20*b^2*d^3*x^3 - 4*a*b*d^3 + 120*b^2*d*x)*cos(d*x + c) + 2*(5*b^2*d^4*x^4 + 4*a*b*d^4*x - 60*b^
2*d^2*x^2 + 120*b^2)*sin(d*x + c))/d^6

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.34 (sec) , antiderivative size = 921, normalized size of antiderivative = 5.72 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\text {Too large to display} \]

[In]

integrate((b*x^3+a)^2*sin(d*x+c)/x,x, algorithm="giac")

[Out]

1/2*(2*b^2*d^5*x^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*b^2*d^5*x^5*tan(1/2*d*x + 1/2*c)^2 - 2*b^2*d^5*x^5*
tan(1/2*c)^2 + 20*b^2*d^4*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 + 4*a*b*d^5*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2
*c)^2 - a^2*d^6*imag_part(cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^2*d^6*imag_part(cos_integ
ral(-d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^2*d^6*sin_integral(d*x)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)
^2 - 2*b^2*d^5*x^5 + 2*a^2*d^6*real_part(cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 2*a^2*d^6*real
_part(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 40*b^2*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)
^2 + 20*b^2*d^4*x^4*tan(1/2*d*x + 1/2*c) + 4*a*b*d^5*x^2*tan(1/2*d*x + 1/2*c)^2 + a^2*d^6*imag_part(cos_integr
al(d*x))*tan(1/2*d*x + 1/2*c)^2 - a^2*d^6*imag_part(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^6*sin
_integral(d*x)*tan(1/2*d*x + 1/2*c)^2 - 4*a*b*d^5*x^2*tan(1/2*c)^2 - a^2*d^6*imag_part(cos_integral(d*x))*tan(
1/2*c)^2 + a^2*d^6*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 - 2*a^2*d^6*sin_integral(d*x)*tan(1/2*c)^2 - 40*
b^2*d^3*x^3*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^6*real_part(cos_integral(d*x))*tan(1/2*c) + 2*a^2*d^6*real_part(c
os_integral(-d*x))*tan(1/2*c) + 40*b^2*d^3*x^3*tan(1/2*c)^2 + 16*a*b*d^4*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 -
4*a*b*d^5*x^2 + a^2*d^6*imag_part(cos_integral(d*x)) - a^2*d^6*imag_part(cos_integral(-d*x)) + 2*a^2*d^6*sin_
integral(d*x) - 240*b^2*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 8*a*b*d^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c
)^2 + 40*b^2*d^3*x^3 + 16*a*b*d^4*x*tan(1/2*d*x + 1/2*c) + 240*b^2*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2
40*b^2*d^2*x^2*tan(1/2*d*x + 1/2*c) - 8*a*b*d^3*tan(1/2*d*x + 1/2*c)^2 + 8*a*b*d^3*tan(1/2*c)^2 + 240*b^2*d*x*
tan(1/2*d*x + 1/2*c)^2 - 240*b^2*d*x*tan(1/2*c)^2 + 8*a*b*d^3 + 480*b^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 24
0*b^2*d*x + 480*b^2*tan(1/2*d*x + 1/2*c))/(d^6*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + d^6*tan(1/2*d*x + 1/2*c)^
2 + d^6*tan(1/2*c)^2 + d^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x} \,d x \]

[In]

int((sin(c + d*x)*(a + b*x^3)^2)/x,x)

[Out]

int((sin(c + d*x)*(a + b*x^3)^2)/x, x)