Integrand size = 19, antiderivative size = 161 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\frac {4 a b \cos (c+d x)}{d^3}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {120 b^2 \sin (c+d x)}{d^6}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x) \]
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)*s in(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
Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=-\frac {b \left (2 a d^2 \left (-2+d^2 x^2\right )+b x \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \cos (c+d x)}{d^5}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {b \left (4 a d^4 x+5 b \left (24-12 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^6}+a^2 \cos (c) \text {Si}(d x) \]
-((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]
Time = 0.43 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3820, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{x}+2 a b x^2 \sin (c+d x)+b^2 x^5 \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 \sin (c) \operatorname {CosIntegral}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {4 a b \cos (c+d x)}{d^3}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {b^2 x^5 \cos (c+d x)}{d}\) |
(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]
3.1.89.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_ )], x_Symbol] :> Int[ExpandIntegrand[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]
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 | \(a^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )-\frac {6 a b \,c^{2} \cos \left (d x +c \right )}{d^{3}}-\frac {6 a b c \left (c +1\right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {2 \left (c^{2}+c +1\right ) a b \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {6 b^{2} c^{5} \cos \left (d x +c \right )}{d^{6}}+\frac {15 \left (c +1\right ) b^{2} c^{4} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {20 b^{2} c^{3} \left (c^{2}+c +1\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {15 \left (c^{3}+c^{2}+c +1\right ) b^{2} c^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}-\frac {6 b^{2} c \left (c^{4}+c^{3}+c^{2}+c +1\right ) \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {\left (c^{5}+c^{4}+c^{3}+c^{2}+c +1\right ) b^{2} \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}\) | \(487\) |
-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
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\frac {a^{2} d^{6} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + a^{2} d^{6} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \cos \left (d x + c\right ) + {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x - 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{6}} \]
(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
Time = 3.69 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=a^{2} \sin {\left (c \right )} \operatorname {Ci}{\left (d x \right )} + a^{2} \cos {\left (c \right )} \operatorname {Si}{\left (d x \right )} + 2 a b x^{2} \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 4 a b \left (\begin {cases} \frac {x^{3} \sin {\left (c \right )}}{3} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x \sin {\left (c + d x \right )}}{d} + \frac {\cos {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cos {\left (c \right )}}{2} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) + b^{2} x^{5} \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 5 b^{2} \left (\begin {cases} \frac {x^{6} \sin {\left (c \right )}}{6} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x^{4} \sin {\left (c + d x \right )}}{d} + \frac {4 x^{3} \cos {\left (c + d x \right )}}{d^{2}} - \frac {12 x^{2} \sin {\left (c + d x \right )}}{d^{3}} - \frac {24 x \cos {\left (c + d x \right )}}{d^{4}} + \frac {24 \sin {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\frac {x^{5} \cos {\left (c \right )}}{5} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \]
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)), (-P iecewise((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))
Result contains complex when optimal does not.
Time = 5.68 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\frac {{\left (a^{2} {\left (-i \, {\rm Ei}\left (i \, d x\right ) + i \, {\rm Ei}\left (-i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left ({\rm Ei}\left (i \, d x\right ) + {\rm Ei}\left (-i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 2 \, {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x - 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{6}} \]
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
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 \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\text {Too large to display} \]
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_integral(-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 + 2 0*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_integral(d*x))*tan(1/2*d*x + 1/2*c)^2 - a^2*d^6*im ag_part(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^6*sin_integra l(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(cos_integral(-d*x))*tan(1/2*c) + 40*b^2*d^3*x^3*t an(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...
Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x} \,d x \]