\(\int \frac {(a+b x^3)^2 \sin (c+d x)}{x^2} \, dx\) [90]

Optimal result

Integrand size = 19, antiderivative size = 145 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^2} \, dx=-\frac {24 b^2 \cos (c+d x)}{d^5}-\frac {2 a b x \cos (c+d x)}{d}+\frac {12 b^2 x^2 \cos (c+d x)}{d^3}-\frac {b^2 x^4 \cos (c+d x)}{d}+a^2 d \cos (c) \operatorname {CosIntegral}(d x)+\frac {2 a b \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{x}-\frac {24 b^2 x \sin (c+d x)}{d^4}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-a^2 d \sin (c) \text {Si}(d x) \]

[Out]

a^2*d*Ci(d*x)*cos(c)-24*b^2*cos(d*x+c)/d^5-2*a*b*x*cos(d*x+c)/d+12*b^2*x^2*cos(d*x+c)/d^3-b^2*x^4*cos(d*x+c)/d
-a^2*d*Si(d*x)*sin(c)+2*a*b*sin(d*x+c)/d^2-a^2*sin(d*x+c)/x-24*b^2*x*sin(d*x+c)/d^4+4*b^2*x^3*sin(d*x+c)/d^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3420, 3378, 3384, 3380, 3383, 3377, 2717, 2718} \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^2} \, dx=a^2 d \cos (c) \operatorname {CosIntegral}(d x)-a^2 d \sin (c) \text {Si}(d x)-\frac {a^2 \sin (c+d x)}{x}+\frac {2 a b \sin (c+d x)}{d^2}-\frac {2 a b x \cos (c+d x)}{d}-\frac {24 b^2 \cos (c+d x)}{d^5}-\frac {24 b^2 x \sin (c+d x)}{d^4}+\frac {12 b^2 x^2 \cos (c+d x)}{d^3}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-\frac {b^2 x^4 \cos (c+d x)}{d} \]

[In]

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

[Out]

(-24*b^2*Cos[c + d*x])/d^5 - (2*a*b*x*Cos[c + d*x])/d + (12*b^2*x^2*Cos[c + d*x])/d^3 - (b^2*x^4*Cos[c + d*x])
/d + a^2*d*Cos[c]*CosIntegral[d*x] + (2*a*b*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/x - (24*b^2*x*Sin[c + d*x])
/d^4 + (4*b^2*x^3*Sin[c + d*x])/d^2 - a^2*d*Sin[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 3378

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

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}+2 a b x \sin (c+d x)+b^2 x^4 \sin (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\sin (c+d x)}{x^2} \, dx+(2 a b) \int x \sin (c+d x) \, dx+b^2 \int x^4 \sin (c+d x) \, dx \\ & = -\frac {2 a b x \cos (c+d x)}{d}-\frac {b^2 x^4 \cos (c+d x)}{d}-\frac {a^2 \sin (c+d x)}{x}+\frac {(2 a b) \int \cos (c+d x) \, dx}{d}+\frac {\left (4 b^2\right ) \int x^3 \cos (c+d x) \, dx}{d}+\left (a^2 d\right ) \int \frac {\cos (c+d x)}{x} \, dx \\ & = -\frac {2 a b x \cos (c+d x)}{d}-\frac {b^2 x^4 \cos (c+d x)}{d}+\frac {2 a b \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{x}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-\frac {\left (12 b^2\right ) \int x^2 \sin (c+d x) \, dx}{d^2}+\left (a^2 d \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx-\left (a^2 d \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx \\ & = -\frac {2 a b x \cos (c+d x)}{d}+\frac {12 b^2 x^2 \cos (c+d x)}{d^3}-\frac {b^2 x^4 \cos (c+d x)}{d}+a^2 d \cos (c) \operatorname {CosIntegral}(d x)+\frac {2 a b \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{x}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-a^2 d \sin (c) \text {Si}(d x)-\frac {\left (24 b^2\right ) \int x \cos (c+d x) \, dx}{d^3} \\ & = -\frac {2 a b x \cos (c+d x)}{d}+\frac {12 b^2 x^2 \cos (c+d x)}{d^3}-\frac {b^2 x^4 \cos (c+d x)}{d}+a^2 d \cos (c) \operatorname {CosIntegral}(d x)+\frac {2 a b \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{x}-\frac {24 b^2 x \sin (c+d x)}{d^4}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-a^2 d \sin (c) \text {Si}(d x)+\frac {\left (24 b^2\right ) \int \sin (c+d x) \, dx}{d^4} \\ & = -\frac {24 b^2 \cos (c+d x)}{d^5}-\frac {2 a b x \cos (c+d x)}{d}+\frac {12 b^2 x^2 \cos (c+d x)}{d^3}-\frac {b^2 x^4 \cos (c+d x)}{d}+a^2 d \cos (c) \operatorname {CosIntegral}(d x)+\frac {2 a b \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{x}-\frac {24 b^2 x \sin (c+d x)}{d^4}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-a^2 d \sin (c) \text {Si}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^2} \, dx=-\frac {24 b^2 \cos (c+d x)}{d^5}-\frac {2 a b x \cos (c+d x)}{d}+\frac {12 b^2 x^2 \cos (c+d x)}{d^3}-\frac {b^2 x^4 \cos (c+d x)}{d}+a^2 d \cos (c) \operatorname {CosIntegral}(d x)+\frac {2 a b \sin (c+d x)}{d^2}-\frac {a^2 \sin (c+d x)}{x}-\frac {24 b^2 x \sin (c+d x)}{d^4}+\frac {4 b^2 x^3 \sin (c+d x)}{d^2}-a^2 d \sin (c) \text {Si}(d x) \]

[In]

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

[Out]

(-24*b^2*Cos[c + d*x])/d^5 - (2*a*b*x*Cos[c + d*x])/d + (12*b^2*x^2*Cos[c + d*x])/d^3 - (b^2*x^4*Cos[c + d*x])
/d + a^2*d*Cos[c]*CosIntegral[d*x] + (2*a*b*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/x - (24*b^2*x*Sin[c + d*x])
/d^4 + (4*b^2*x^3*Sin[c + d*x])/d^2 - a^2*d*Sin[c]*SinIntegral[d*x]

Maple [C] (warning: unable to verify)

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

Time = 0.49 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.48

method	result	size
risch	\(-\frac {-\pi \,\operatorname {csgn}\left (d x \right ) \sin \left (c \right ) a^{2} d^{6} x +2 \,\operatorname {Si}\left (d x \right ) \sin \left (c \right ) a^{2} d^{6} x -i \pi \,\operatorname {csgn}\left (d x \right ) \cos \left (c \right ) a^{2} d^{6} x +2 \cos \left (d x +c \right ) b^{2} d^{4} x^{5}+2 i \operatorname {Si}\left (d x \right ) \cos \left (c \right ) a^{2} d^{6} x +2 \,\operatorname {Ei}_{1}\left (-i d x \right ) \cos \left (c \right ) a^{2} d^{6} x -8 \sin \left (d x +c \right ) b^{2} d^{3} x^{4}+4 \cos \left (d x +c \right ) a b \,d^{4} x^{2}+2 \sin \left (d x +c \right ) a^{2} d^{5}-24 \cos \left (d x +c \right ) b^{2} d^{2} x^{3}-4 \sin \left (d x +c \right ) a b \,d^{3} x +48 \sin \left (d x +c \right ) b^{2} d \,x^{2}+48 \cos \left (d x +c \right ) b^{2} x}{2 x \,d^{5}}\)	\(214\)
meijerg	\(\frac {16 b^{2} \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {x \left (d^{2}\right )^{\frac {5}{2}} \left (-\frac {5 d^{2} x^{2}}{2}+15\right ) \cos \left (d x \right )}{10 \sqrt {\pi }\, d^{4}}+\frac {\left (d^{2}\right )^{\frac {5}{2}} \left (\frac {5}{8} d^{4} x^{4}-\frac {15}{2} d^{2} x^{2}+15\right ) \sin \left (d x \right )}{10 \sqrt {\pi }\, d^{5}}\right )}{d^{4} \sqrt {d^{2}}}+\frac {16 b^{2} \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}-\frac {9}{2} d^{2} x^{2}+9\right ) \cos \left (d x \right )}{6 \sqrt {\pi }}-\frac {d x \left (-\frac {3 d^{2} x^{2}}{2}+9\right ) \sin \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {4 a b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {d x \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {4 a b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {d x \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a^{2} \sqrt {\pi }\, \sin \left (c \right ) d^{2} \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{4 \sqrt {d^{2}}}+\frac {a^{2} \sqrt {\pi }\, \cos \left (c \right ) d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}+\frac {4 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{4}\)	\(360\)
derivativedivides	\(d \left (-\frac {15 b^{2} c^{4} \cos \left (d x +c \right )}{d^{6}}+a^{2} \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )+\frac {15 \left (3 c^{2}+2 c +1\right ) c^{2} b^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {2 \left (2 c +1\right ) a b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {20 b^{2} c^{3} \left (2 c +1\right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {6 c \,b^{2} \left (4 c^{3}+3 c^{2}+2 c +1\right ) \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}+\frac {6 a b c \cos \left (d x +c \right )}{d^{3}}+\frac {\left (5 c^{4}+4 c^{3}+3 c^{2}+2 c +1\right ) b^{2} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}\right )\)	\(365\)
default	\(d \left (-\frac {15 b^{2} c^{4} \cos \left (d x +c \right )}{d^{6}}+a^{2} \left (-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )\right )+\frac {15 \left (3 c^{2}+2 c +1\right ) c^{2} b^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {2 \left (2 c +1\right ) a b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {20 b^{2} c^{3} \left (2 c +1\right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {6 c \,b^{2} \left (4 c^{3}+3 c^{2}+2 c +1\right ) \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}+\frac {6 a b c \cos \left (d x +c \right )}{d^{3}}+\frac {\left (5 c^{4}+4 c^{3}+3 c^{2}+2 c +1\right ) b^{2} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}\right )\)	\(365\)

[In]

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

[Out]

-1/2/x/d^5*(-Pi*csgn(d*x)*sin(c)*a^2*d^6*x+2*Si(d*x)*sin(c)*a^2*d^6*x-I*Pi*csgn(d*x)*cos(c)*a^2*d^6*x+2*cos(d*
x+c)*b^2*d^4*x^5+2*I*Si(d*x)*cos(c)*a^2*d^6*x+2*Ei(1,-I*d*x)*cos(c)*a^2*d^6*x-8*sin(d*x+c)*b^2*d^3*x^4+4*cos(d
*x+c)*a*b*d^4*x^2+2*sin(d*x+c)*a^2*d^5-24*cos(d*x+c)*b^2*d^2*x^3-4*sin(d*x+c)*a*b*d^3*x+48*sin(d*x+c)*b^2*d*x^
2+48*cos(d*x+c)*b^2*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^2} \, dx=\frac {a^{2} d^{6} x \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) - a^{2} d^{6} x \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - {\left (b^{2} d^{4} x^{5} + 2 \, a b d^{4} x^{2} - 12 \, b^{2} d^{2} x^{3} + 24 \, b^{2} x\right )} \cos \left (d x + c\right ) + {\left (4 \, b^{2} d^{3} x^{4} - a^{2} d^{5} + 2 \, a b d^{3} x - 24 \, b^{2} d x^{2}\right )} \sin \left (d x + c\right )}{d^{5} x} \]

[In]

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

[Out]

(a^2*d^6*x*cos(c)*cos_integral(d*x) - a^2*d^6*x*sin(c)*sin_integral(d*x) - (b^2*d^4*x^5 + 2*a*b*d^4*x^2 - 12*b
^2*d^2*x^3 + 24*b^2*x)*cos(d*x + c) + (4*b^2*d^3*x^4 - a^2*d^5 + 2*a*b*d^3*x - 24*b^2*d*x^2)*sin(d*x + c))/(d^
5*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.47 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^2} \, dx=\frac {{\left (a^{2} {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left (-i \, \Gamma \left (-1, i \, d x\right ) + i \, \Gamma \left (-1, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 2 \, {\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x - 12 \, b^{2} d^{2} x^{2} + 24 \, b^{2}\right )} \cos \left (d x + c\right ) + 4 \, {\left (2 \, b^{2} d^{3} x^{3} + a b d^{3} - 12 \, b^{2} d x\right )} \sin \left (d x + c\right )}{2 \, d^{5}} \]

[In]

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

[Out]

1/2*((a^2*(gamma(-1, I*d*x) + gamma(-1, -I*d*x))*cos(c) + a^2*(-I*gamma(-1, I*d*x) + I*gamma(-1, -I*d*x))*sin(
c))*d^6 - 2*(b^2*d^4*x^4 + 2*a*b*d^4*x - 12*b^2*d^2*x^2 + 24*b^2)*cos(d*x + c) + 4*(2*b^2*d^3*x^3 + a*b*d^3 -
12*b^2*d*x)*sin(d*x + c))/d^5

Giac [C] (verification not implemented)

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

Time = 0.34 (sec) , antiderivative size = 2038, normalized size of antiderivative = 14.06 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(2*b^2*d^4*x^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^6*x*real_part(cos_integral(d*x))
*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^6*x*real_part(cos_integral(-d*x))*tan(1/2*d*x + 1/
2*c)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*b^2*d^4*x^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2 - 2*a^2*d^6*x*imag_pa
rt(cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^6*x*imag_part(cos_integral(-d
*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c) - 4*a^2*d^6*x*sin_integral(d*x)*tan(1/2*d*x + 1/2*c)^2*t
an(1/2*d*x)^2*tan(1/2*c) + 2*b^2*d^4*x^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*b^2*d^4*x^5*tan(1/2*d*x)^2*ta
n(1/2*c)^2 + a^2*d^6*x*real_part(cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2 + a^2*d^6*x*real_par
t(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2 - a^2*d^6*x*real_part(cos_integral(d*x))*tan(1/2*d
*x + 1/2*c)^2*tan(1/2*c)^2 - a^2*d^6*x*real_part(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - a^2
*d^6*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^6*x*real_part(cos_integral(-d*x))*tan(
1/2*d*x)^2*tan(1/2*c)^2 + 16*b^2*d^3*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*a*b*d^4*x^2*tan(
1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*b^2*d^4*x^5*tan(1/2*d*x + 1/2*c)^2 - 2*b^2*d^4*x^5*tan(1/2*
d*x)^2 - 2*a^2*d^6*x*imag_part(cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 2*a^2*d^6*x*imag_part(co
s_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 4*a^2*d^6*x*sin_integral(d*x)*tan(1/2*d*x + 1/2*c)^2*tan
(1/2*c) - 2*a^2*d^6*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^6*x*imag_part(cos_integ
ral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a^2*d^6*x*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c) - 2*b^2*d^4*x^5
*tan(1/2*c)^2 - 24*b^2*d^2*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*d^6*x*real_part(cos_in
tegral(d*x))*tan(1/2*d*x + 1/2*c)^2 + a^2*d^6*x*real_part(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2 + a^2*d^6
*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^6*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 + 16*b
^2*d^3*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x)^2 + 4*a*b*d^4*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2 + 4*a^2*d
^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c) - a^2*d^6*x*real_part(cos_integral(d*x))*tan(1/2*c)^2 - a^
2*d^6*x*real_part(cos_integral(-d*x))*tan(1/2*c)^2 + 16*b^2*d^3*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 + 4*a*b*
d^4*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 4*a^2*d^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)*tan(1/2*c)^2 - 4*a
*b*d^4*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*b^2*d^4*x^5 - 24*b^2*d^2*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2
- 2*a^2*d^6*x*imag_part(cos_integral(d*x))*tan(1/2*c) + 2*a^2*d^6*x*imag_part(cos_integral(-d*x))*tan(1/2*c) -
4*a^2*d^6*x*sin_integral(d*x)*tan(1/2*c) - 24*b^2*d^2*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 24*b^2*d^2*x^
3*tan(1/2*d*x)^2*tan(1/2*c)^2 + 8*a*b*d^3*x*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*d^6*x*real_
part(cos_integral(d*x)) + a^2*d^6*x*real_part(cos_integral(-d*x)) + 16*b^2*d^3*x^4*tan(1/2*d*x + 1/2*c) + 4*a*
b*d^4*x^2*tan(1/2*d*x + 1/2*c)^2 - 4*a^2*d^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x) - 4*a*b*d^4*x^2*tan(1/2*d*x)^
2 - 4*a^2*d^5*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 4*a^2*d^5*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*b*d^4*x^2*tan(1/2*
c)^2 + 4*a^2*d^5*tan(1/2*d*x)*tan(1/2*c)^2 - 96*b^2*d*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2
4*b^2*d^2*x^3*tan(1/2*d*x + 1/2*c)^2 + 24*b^2*d^2*x^3*tan(1/2*d*x)^2 + 8*a*b*d^3*x*tan(1/2*d*x + 1/2*c)*tan(1/
2*d*x)^2 + 24*b^2*d^2*x^3*tan(1/2*c)^2 + 8*a*b*d^3*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 + 48*b^2*x*tan(1/2*d*x
+ 1/2*c)^2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^4*x^2 - 4*a^2*d^5*tan(1/2*d*x) - 96*b^2*d*x^2*tan(1/2*d*x + 1
/2*c)*tan(1/2*d*x)^2 - 4*a^2*d^5*tan(1/2*c) - 96*b^2*d*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 + 24*b^2*d^2*x^3
+ 8*a*b*d^3*x*tan(1/2*d*x + 1/2*c) + 48*b^2*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2 + 48*b^2*x*tan(1/2*d*x + 1
/2*c)^2*tan(1/2*c)^2 - 48*b^2*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 96*b^2*d*x^2*tan(1/2*d*x + 1/2*c) + 48*b^2*x*tan
(1/2*d*x + 1/2*c)^2 - 48*b^2*x*tan(1/2*d*x)^2 - 48*b^2*x*tan(1/2*c)^2 - 48*b^2*x)/(d^5*x*tan(1/2*d*x + 1/2*c)^
2*tan(1/2*d*x)^2*tan(1/2*c)^2 + d^5*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x)^2 + d^5*x*tan(1/2*d*x + 1/2*c)^2*tan
(1/2*c)^2 + d^5*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + d^5*x*tan(1/2*d*x + 1/2*c)^2 + d^5*x*tan(1/2*d*x)^2 + d^5*x*ta
n(1/2*c)^2 + d^5*x)

Mupad [F(-1)]

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

[In]

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

[Out]

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