Integrand size = 19, antiderivative size = 111 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=\frac {6 b^2 x \cos (c+d x)}{d^3}-\frac {2 a b x \cos (c+d x)}{d}-\frac {b^2 x^3 \cos (c+d x)}{d}+a^2 \operatorname {CosIntegral}(d x) \sin (c)-\frac {6 b^2 \sin (c+d x)}{d^4}+\frac {2 a b \sin (c+d x)}{d^2}+\frac {3 b^2 x^2 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x) \] Output:
6*b^2*x*cos(d*x+c)/d^3-2*a*b*x*cos(d*x+c)/d-b^2*x^3*cos(d*x+c)/d+a^2*Ci(d* x)*sin(c)-6*b^2*sin(d*x+c)/d^4+2*a*b*sin(d*x+c)/d^2+3*b^2*x^2*sin(d*x+c)/d ^2+a^2*cos(c)*Si(d*x)
Time = 0.43 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=-\frac {b x \left (2 a d^2+b \left (-6+d^2 x^2\right )\right ) \cos (c+d x)}{d^3}+a^2 \operatorname {CosIntegral}(d x) \sin (c)+\frac {b \left (2 a d^2+3 b \left (-2+d^2 x^2\right )\right ) \sin (c+d x)}{d^4}+a^2 \cos (c) \text {Si}(d x) \] Input:
Integrate[((a + b*x^2)^2*Sin[c + d*x])/x,x]
Output:
-((b*x*(2*a*d^2 + b*(-6 + d^2*x^2))*Cos[c + d*x])/d^3) + a^2*CosIntegral[d *x]*Sin[c] + (b*(2*a*d^2 + 3*b*(-2 + d^2*x^2))*Sin[c + d*x])/d^4 + a^2*Cos [c]*SinIntegral[d*x]
Time = 0.39 (sec) , antiderivative size = 111, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{x}+2 a b x \sin (c+d x)+b^2 x^3 \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 \sin (c) \operatorname {CosIntegral}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {2 a b \sin (c+d x)}{d^2}-\frac {2 a b x \cos (c+d x)}{d}-\frac {6 b^2 \sin (c+d x)}{d^4}+\frac {6 b^2 x \cos (c+d x)}{d^3}+\frac {3 b^2 x^2 \sin (c+d x)}{d^2}-\frac {b^2 x^3 \cos (c+d x)}{d}\) |
Input:
Int[((a + b*x^2)^2*Sin[c + d*x])/x,x]
Output:
(6*b^2*x*Cos[c + d*x])/d^3 - (2*a*b*x*Cos[c + d*x])/d - (b^2*x^3*Cos[c + d *x])/d + a^2*CosIntegral[d*x]*Sin[c] - (6*b^2*Sin[c + d*x])/d^4 + (2*a*b*S in[c + d*x])/d^2 + (3*b^2*x^2*Sin[c + d*x])/d^2 + a^2*Cos[c]*SinIntegral[d *x]
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 = 1.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {b^{2} x^{3} \cos \left (d x +c \right )}{d}-\frac {i a^{2} {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (i d x \right )}{2}+\frac {i a^{2} {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2}+\frac {3 b^{2} x^{2} \sin \left (d x +c \right )}{d^{2}}-\frac {2 a b x \cos \left (d x +c \right )}{d}+\frac {2 a b \sin \left (d x +c \right )}{d^{2}}+\frac {6 b^{2} x \cos \left (d x +c \right )}{d^{3}}-\frac {6 b^{2} \sin \left (d x +c \right )}{d^{4}}\) | \(128\) |
| derivativedivides | \(a^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )+\frac {4 a b c \cos \left (d x +c \right )}{d^{2}}+\frac {2 \left (c +1\right ) a b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\frac {4 c^{3} b^{2} \cos \left (d x +c \right )}{d^{4}}+\frac {6 \left (c +1\right ) b^{2} c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}-\frac {4 b^{2} c \left (c^{2}+c +1\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}+\frac {\left (c^{3}+c^{2}+c +1\right ) b^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}\) | \(236\) |
| default | \(a^{2} \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )+\frac {4 a b c \cos \left (d x +c \right )}{d^{2}}+\frac {2 \left (c +1\right ) a b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\frac {4 c^{3} b^{2} \cos \left (d x +c \right )}{d^{4}}+\frac {6 \left (c +1\right ) b^{2} c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}-\frac {4 b^{2} c \left (c^{2}+c +1\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}+\frac {\left (c^{3}+c^{2}+c +1\right ) b^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}\) | \(236\) |
| meijerg | \(\frac {8 b^{2} \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 x^{2} d^{2}}{2}+3\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}-\frac {x d \left (-\frac {x^{2} d^{2}}{2}+3\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}+\frac {8 b^{2} \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {x d \left (-\frac {5 x^{2} d^{2}}{2}+15\right ) \cos \left (d x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 x^{2} d^{2}}{2}+15\right ) \sin \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {4 a b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {x d \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {4 a b \cos \left (c \right ) \sqrt {\pi }\, \left (-\frac {d x \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a^{2} \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+a^{2} \cos \left (c \right ) \operatorname {Si}\left (d x \right )\) | \(255\) |
Input:
int((b*x^2+a)^2*sin(d*x+c)/x,x,method=_RETURNVERBOSE)
Output:
-b^2*x^3*cos(d*x+c)/d-1/2*I*a^2*exp(-I*c)*Ei(1,I*d*x)+1/2*I*a^2*exp(I*c)*E i(1,-I*d*x)+3*b^2*x^2*sin(d*x+c)/d^2-2*a*b*x*cos(d*x+c)/d+2*a*b*sin(d*x+c) /d^2+6*b^2*x*cos(d*x+c)/d^3-6*b^2*sin(d*x+c)/d^4
Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=\frac {a^{2} d^{4} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + a^{2} d^{4} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - {\left (b^{2} d^{3} x^{3} + 2 \, {\left (a b d^{3} - 3 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) + {\left (3 \, b^{2} d^{2} x^{2} + 2 \, a b d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{4}} \] Input:
integrate((b*x^2+a)^2*sin(d*x+c)/x,x, algorithm="fricas")
Output:
(a^2*d^4*cos_integral(d*x)*sin(c) + a^2*d^4*cos(c)*sin_integral(d*x) - (b^ 2*d^3*x^3 + 2*(a*b*d^3 - 3*b^2*d)*x)*cos(d*x + c) + (3*b^2*d^2*x^2 + 2*a*b *d^2 - 6*b^2)*sin(d*x + c))/d^4
Time = 2.38 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=a^{2} \sin {\left (c \right )} \operatorname {Ci}{\left (d x \right )} + a^{2} \cos {\left (c \right )} \operatorname {Si}{\left (d x \right )} + 2 a b x \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 2 a b \left (\begin {cases} \frac {x^{2} \sin {\left (c \right )}}{2} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\left (c \right )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) + b^{2} x^{3} \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 3 b^{2} \left (\begin {cases} \frac {x^{4} \sin {\left (c \right )}}{4} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x^{2} \sin {\left (c + d x \right )}}{d} + \frac {2 x \cos {\left (c + d x \right )}}{d^{2}} - \frac {2 \sin {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{3} \cos {\left (c \right )}}{3} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((b*x**2+a)**2*sin(d*x+c)/x,x)
Output:
a**2*sin(c)*Ci(d*x) + a**2*cos(c)*Si(d*x) + 2*a*b*x*Piecewise((x*sin(c), E q(d, 0)), (-cos(c + d*x)/d, True)) - 2*a*b*Piecewise((x**2*sin(c)/2, Eq(d, 0)), (-Piecewise((sin(c + d*x)/d, Ne(d, 0)), (x*cos(c), True))/d, True)) + b**2*x**3*Piecewise((x*sin(c), Eq(d, 0)), (-cos(c + d*x)/d, True)) - 3*b **2*Piecewise((x**4*sin(c)/4, Eq(d, 0)), (-Piecewise((x**2*sin(c + d*x)/d + 2*x*cos(c + d*x)/d**2 - 2*sin(c + d*x)/d**3, Ne(d, 0)), (x**3*cos(c)/3, True))/d, True))
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=\frac {{\left (a^{2} {\left (-i \, {\rm Ei}\left (i \, d x\right ) + i \, {\rm Ei}\left (-i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left ({\rm Ei}\left (i \, d x\right ) + {\rm Ei}\left (-i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4} - 2 \, {\left (b^{2} d^{3} x^{3} + 2 \, {\left (a b d^{3} - 3 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 2 \, a b d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{4}} \] Input:
integrate((b*x^2+a)^2*sin(d*x+c)/x,x, algorithm="maxima")
Output:
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^4 - 2*(b^2*d^3*x^3 + 2*(a*b*d^3 - 3*b^2*d)*x)*cos(d*x + c) + 2*(3*b^2*d^2*x^2 + 2*a*b*d^2 - 6*b^2)*sin(d*x + c))/d^4
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 725, normalized size of antiderivative = 6.53 \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^2*sin(d*x+c)/x,x, algorithm="giac")
Output:
1/2*(2*b^2*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - a^2*d^4*imag_part (cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^2*d^4*imag_par t(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^2*d^4*sin_ integral(d*x)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*b^2*d^3*x^3*tan(1/2* d*x + 1/2*c)^2 + 2*a^2*d^4*real_part(cos_integral(d*x))*tan(1/2*d*x + 1/2* c)^2*tan(1/2*c) + 2*a^2*d^4*real_part(cos_integral(-d*x))*tan(1/2*d*x + 1/ 2*c)^2*tan(1/2*c) - 2*b^2*d^3*x^3*tan(1/2*c)^2 + 4*a*b*d^3*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^2*d^4*imag_part(cos_integral(d*x))*tan(1/2*d*x + 1/2*c)^2 - a^2*d^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^4*sin_integral(d*x)*tan(1/2*d*x + 1/2*c)^2 - a^2*d^4*imag_part(c os_integral(d*x))*tan(1/2*c)^2 + a^2*d^4*imag_part(cos_integral(-d*x))*tan (1/2*c)^2 - 2*a^2*d^4*sin_integral(d*x)*tan(1/2*c)^2 + 12*b^2*d^2*x^2*tan( 1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 2*b^2*d^3*x^3 + 4*a*b*d^3*x*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^4*real_part(cos_integral(d*x))*tan(1/2*c) + 2*a^2*d^4*r eal_part(cos_integral(-d*x))*tan(1/2*c) - 4*a*b*d^3*x*tan(1/2*c)^2 - 12*b^ 2*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^2*d^4*imag_part(cos_integral (d*x)) - a^2*d^4*imag_part(cos_integral(-d*x)) + 2*a^2*d^4*sin_integral(d* x) + 12*b^2*d^2*x^2*tan(1/2*d*x + 1/2*c) + 8*a*b*d^2*tan(1/2*d*x + 1/2*c)* tan(1/2*c)^2 - 4*a*b*d^3*x - 12*b^2*d*x*tan(1/2*d*x + 1/2*c)^2 + 12*b^2*d* x*tan(1/2*c)^2 + 8*a*b*d^2*tan(1/2*d*x + 1/2*c) - 24*b^2*tan(1/2*d*x + ...
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x} \,d x \] Input:
int((sin(c + d*x)*(a + b*x^2)^2)/x,x)
Output:
int((sin(c + d*x)*(a + b*x^2)^2)/x, x)
\[ \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x} \, dx=\frac {-2 \cos \left (d x +c \right ) a b \,d^{3} x -\cos \left (d x +c \right ) b^{2} d^{3} x^{3}+6 \cos \left (d x +c \right ) b^{2} d x +\left (\int \frac {\sin \left (d x +c \right )}{x}d x \right ) a^{2} d^{4}+2 \sin \left (d x +c \right ) a b \,d^{2}+3 \sin \left (d x +c \right ) b^{2} d^{2} x^{2}-6 \sin \left (d x +c \right ) b^{2}}{d^{4}} \] Input:
int((b*x^2+a)^2*sin(d*x+c)/x,x)
Output:
( - 2*cos(c + d*x)*a*b*d**3*x - cos(c + d*x)*b**2*d**3*x**3 + 6*cos(c + d* x)*b**2*d*x + int(sin(c + d*x)/x,x)*a**2*d**4 + 2*sin(c + d*x)*a*b*d**2 + 3*sin(c + d*x)*b**2*d**2*x**2 - 6*sin(c + d*x)*b**2)/d**4