Integrand size = 17, antiderivative size = 41 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=-\frac {b x \cos (c+d x)}{d}+a \operatorname {CosIntegral}(d x) \sin (c)+\frac {b \sin (c+d x)}{d^2}+a \cos (c) \text {Si}(d x) \]
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=-\frac {b \cos (d x) (d x \cos (c)-\sin (c))}{d^2}+a \operatorname {CosIntegral}(d x) \sin (c)+\frac {b (\cos (c)+d x \sin (c)) \sin (d x)}{d^2}+a \cos (c) \text {Si}(d x) \]
-((b*Cos[d*x]*(d*x*Cos[c] - Sin[c]))/d^2) + a*CosIntegral[d*x]*Sin[c] + (b *(Cos[c] + d*x*Sin[c])*Sin[d*x])/d^2 + a*Cos[c]*SinIntegral[d*x]
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx\) |
\(\Big \downarrow \) 3820 |
\(\displaystyle \int \left (\frac {a \sin (c+d x)}{x}+b x \sin (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \sin (c) \operatorname {CosIntegral}(d x)+a \cos (c) \text {Si}(d x)+\frac {b \sin (c+d x)}{d^2}-\frac {b x \cos (c+d x)}{d}\) |
-((b*x*Cos[c + d*x])/d) + a*CosIntegral[d*x]*Sin[c] + (b*Sin[c + d*x])/d^2 + a*Cos[c]*SinIntegral[d*x]
3.1.44.3.1 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]
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.46
method | result | size |
derivativedivides | \(a \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )+\frac {2 b c \cos \left (d x +c \right )}{d^{2}}+\frac {\left (c +1\right ) b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}\) | \(60\) |
default | \(a \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )+\frac {2 b c \cos \left (d x +c \right )}{d^{2}}+\frac {\left (c +1\right ) b \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}\) | \(60\) |
risch | \(\frac {i a \,{\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2}-\frac {i \operatorname {Ei}_{1}\left (-i d x \right ) {\mathrm e}^{-i c} a}{2}-\frac {\pi \,\operatorname {csgn}\left (d x \right ) {\mathrm e}^{-i c} a}{2}+\operatorname {Si}\left (d x \right ) {\mathrm e}^{-i c} a -\frac {b x \cos \left (d x +c \right )}{d}+\frac {b \sin \left (d x +c \right )}{d^{2}}\) | \(82\) |
meijerg | \(\frac {2 b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {d x \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {2 b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {d x \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {2 \gamma +2 \ln \left (x \right )+\ln \left (d^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Ci}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+a \cos \left (c \right ) \operatorname {Si}\left (d x \right )\) | \(136\) |
a*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+2/d^2*b*c*cos(d*x+c)+(c+1)/d^2*b*(sin(d* x+c)-cos(d*x+c)*(d*x+c))
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=\frac {a d^{2} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + a d^{2} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - b d x \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{d^{2}} \]
(a*d^2*cos_integral(d*x)*sin(c) + a*d^2*cos(c)*sin_integral(d*x) - b*d*x*c os(d*x + c) + b*sin(d*x + c))/d^2
Time = 1.87 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=a \sin {\left (c \right )} \operatorname {Ci}{\left (d x \right )} + a \cos {\left (c \right )} \operatorname {Si}{\left (d x \right )} + b x \left (\begin {cases} x \sin {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - b \left (\begin {cases} \frac {x^{2} \sin {\left (c \right )}}{2} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\left (c \right )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \]
a*sin(c)*Ci(d*x) + a*cos(c)*Si(d*x) + b*x*Piecewise((x*sin(c), Eq(d, 0)), (-cos(c + d*x)/d, True)) - b*Piecewise((x**2*sin(c)/2, Eq(d, 0)), (-Piecew ise((sin(c + d*x)/d, Ne(d, 0)), (x*cos(c), True))/d, True))
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=-\frac {2 \, b d x \cos \left (d x + c\right ) - {\left (a {\left (-i \, {\rm Ei}\left (i \, d x\right ) + i \, {\rm Ei}\left (-i \, d x\right )\right )} \cos \left (c\right ) + a {\left ({\rm Ei}\left (i \, d x\right ) + {\rm Ei}\left (-i \, d x\right )\right )} \sin \left (c\right )\right )} d^{2} - 2 \, b \sin \left (d x + c\right )}{2 \, d^{2}} \]
-1/2*(2*b*d*x*cos(d*x + c) - (a*(-I*Ei(I*d*x) + I*Ei(-I*d*x))*cos(c) + a*( Ei(I*d*x) + Ei(-I*d*x))*sin(c))*d^2 - 2*b*sin(d*x + c))/d^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 432, normalized size of antiderivative = 10.54 \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=-\frac {a d^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d^{2} \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 2 \, b d x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - 2 \, a d^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, b d x \tan \left (\frac {1}{2} \, d x\right )^{2} - 2 \, a d^{2} \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 8 \, b d x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, b d x \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{2} \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) + a d^{2} \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) - 2 \, a d^{2} \operatorname {Si}\left (d x\right ) + 4 \, b \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, b \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b d x - 4 \, b \tan \left (\frac {1}{2} \, d x\right ) - 4 \, b \tan \left (\frac {1}{2} \, c\right )}{2 \, {\left (d^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} + d^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2}\right )}} \]
-1/2*(a*d^2*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d ^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*d^2*sin _integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*d^2*real_part(cos_integra l(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^2*real_part(cos_integral(-d*x))* tan(1/2*d*x)^2*tan(1/2*c) + 2*b*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^2*im ag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a*d^2*imag_part(cos_integral(- d*x))*tan(1/2*d*x)^2 - 2*a*d^2*sin_integral(d*x)*tan(1/2*d*x)^2 + a*d^2*im ag_part(cos_integral(d*x))*tan(1/2*c)^2 - a*d^2*imag_part(cos_integral(-d* x))*tan(1/2*c)^2 + 2*a*d^2*sin_integral(d*x)*tan(1/2*c)^2 - 2*b*d*x*tan(1/ 2*d*x)^2 - 2*a*d^2*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d^2*real_ part(cos_integral(-d*x))*tan(1/2*c) - 8*b*d*x*tan(1/2*d*x)*tan(1/2*c) - 2* b*d*x*tan(1/2*c)^2 - a*d^2*imag_part(cos_integral(d*x)) + a*d^2*imag_part( cos_integral(-d*x)) - 2*a*d^2*sin_integral(d*x) + 4*b*tan(1/2*d*x)^2*tan(1 /2*c) + 4*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*b*d*x - 4*b*tan(1/2*d*x) - 4*b*t an(1/2*c))/(d^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + d^2*tan(1/2*d*x)^2 + d^2*tan (1/2*c)^2 + d^2)
Timed out. \[ \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x} \, dx=a\,\mathrm {cosint}\left (d\,x\right )\,\sin \left (c\right )+a\,\mathrm {sinint}\left (d\,x\right )\,\cos \left (c\right )+\frac {b\,\left (\sin \left (c+d\,x\right )-d\,x\,\cos \left (c+d\,x\right )\right )}{d^2} \]