\(\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx\) [20]
Optimal result
Integrand size = 17, antiderivative size = 99 \[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\frac {a \cos (c+d x)}{b^2 d}-\frac {x \cos (c+d x)}{b d}+\frac {a^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}+\frac {\sin (c+d x)}{b d^2}+\frac {a^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}
\]
[Out]
a*cos(d*x+c)/b^2/d-x*cos(d*x+c)/b/d+a^2*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^3-a^2*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^3+si
n(d*x+c)/b/d^2
Rubi [A] (verified)
Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6874, 2718, 3377, 2717, 3384,
3380, 3383} \[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\frac {a^2 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a \cos (c+d x)}{b^2 d}+\frac {\sin (c+d x)}{b d^2}-\frac {x \cos (c+d x)}{b d}
\]
[In]
Int[(x^2*Sin[c + d*x])/(a + b*x),x]
[Out]
(a*Cos[c + d*x])/(b^2*d) - (x*Cos[c + d*x])/(b*d) + (a^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^3 + Si
n[c + d*x]/(b*d^2) + (a^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3
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 6874
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps \begin{align*}
\text {integral}& = \int \left (-\frac {a \sin (c+d x)}{b^2}+\frac {x \sin (c+d x)}{b}+\frac {a^2 \sin (c+d x)}{b^2 (a+b x)}\right ) \, dx \\ & = -\frac {a \int \sin (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {\sin (c+d x)}{a+b x} \, dx}{b^2}+\frac {\int x \sin (c+d x) \, dx}{b} \\ & = \frac {a \cos (c+d x)}{b^2 d}-\frac {x \cos (c+d x)}{b d}+\frac {\int \cos (c+d x) \, dx}{b d}+\frac {\left (a^2 \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}+\frac {\left (a^2 \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2} \\ & = \frac {a \cos (c+d x)}{b^2 d}-\frac {x \cos (c+d x)}{b d}+\frac {a^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}+\frac {\sin (c+d x)}{b d^2}+\frac {a^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88
\[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\frac {a^2 d^2 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+b (d (a-b x) \cos (c+d x)+b \sin (c+d x))+a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3 d^2}
\]
[In]
Integrate[(x^2*Sin[c + d*x])/(a + b*x),x]
[Out]
(a^2*d^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*(d*(a - b*x)*Cos[c + d*x] + b*Sin[c + d*x]) + a^2*d^2*C
os[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^3*d^2)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.82
| | |
| method | result | size |
| | |
| risch |
\(-\frac {\left (d x b -d a \right ) \cos \left (d x +c \right )}{d^{2} b^{2}}+\frac {\sin \left (d x +c \right )}{b \,d^{2}}-\frac {i a^{2} \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{3}}+\frac {i a^{2} \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{3}}+\frac {a^{2} \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{3}}+\frac {a^{2} \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{3}}\) |
\(180\) |
| derivativedivides |
\(\frac {c^{2} d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )+\frac {2 \left (d a -c b \right ) c d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}+\frac {2 c d \cos \left (d x +c \right )}{b}+\frac {\left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {d \left (d a -c b -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{2}}}{d^{3}}\) |
\(318\) |
| default |
\(\frac {c^{2} d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )+\frac {2 \left (d a -c b \right ) c d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}+\frac {2 c d \cos \left (d x +c \right )}{b}+\frac {\left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {d \left (d a -c b -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{2}}}{d^{3}}\) |
\(318\) |
| | |
|
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[In]
int(x^2*sin(d*x+c)/(b*x+a),x,method=_RETURNVERBOSE)
[Out]
-1/d^2*(b*d*x-a*d)/b^2*cos(d*x+c)+sin(d*x+c)/b/d^2-1/2*I*a^2/b^3*cos((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)+1/2*I*a^
2/b^3*cos((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)+1/2*a^2/b^3*sin((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)+1/2*a^2/b^3*sin((
a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.09
\[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=-\frac {a^{2} d^{2} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - a^{2} d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) - b^{2} \sin \left (d x + c\right ) + {\left (b^{2} d x - a b d\right )} \cos \left (d x + c\right )}{b^{3} d^{2}}
\]
[In]
integrate(x^2*sin(d*x+c)/(b*x+a),x, algorithm="fricas")
[Out]
-(a^2*d^2*cos_integral((b*d*x + a*d)/b)*sin(-(b*c - a*d)/b) - a^2*d^2*cos(-(b*c - a*d)/b)*sin_integral((b*d*x
+ a*d)/b) - b^2*sin(d*x + c) + (b^2*d*x - a*b*d)*cos(d*x + c))/(b^3*d^2)
Sympy [F]
\[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\int \frac {x^{2} \sin {\left (c + d x \right )}}{a + b x}\, dx
\]
[In]
integrate(x**2*sin(d*x+c)/(b*x+a),x)
[Out]
Integral(x**2*sin(c + d*x)/(a + b*x), x)
Maxima [F]
\[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\int { \frac {x^{2} \sin \left (d x + c\right )}{b x + a} \,d x }
\]
[In]
integrate(x^2*sin(d*x+c)/(b*x+a),x, algorithm="maxima")
[Out]
-1/2*((b*cos(c)^2 + b*sin(c)^2)*d*x^2*cos(d*x + c) - ((a*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_int
egral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2,
-(I*b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b) - (a*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integ
ral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b
*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 - (b*cos(c)^2 + b*sin(c)^2)*x*sin(d*x + c) - (
(a*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(I*exp_
integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b)
- (a*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(exp_inte
gral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*sin(d
*x + c)^2 + ((b*d*x^2*cos(c) + b*x*sin(c))*cos(d*x + c)^2 + (b*d*x^2*cos(c) + b*x*sin(c))*sin(d*x + c)^2)*cos(
d*x + 2*c) - 2*(((a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3)*cos(d*x + c)
^2 + ((a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3)*sin(d*x + c)^2)*integra
te(1/2*x*cos(d*x + c)/(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2), x) - 2*(((a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x
+ (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x + (a^2*b*co
s(c)^2 + a^2*b*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d
^2)*cos(d*x + c)^2 + (b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sin(d*x + c)^2), x) + ((b*d*x^2*sin(c) - b*x*cos(c)
)*cos(d*x + c)^2 + (b*d*x^2*sin(c) - b*x*cos(c))*sin(d*x + c)^2)*sin(d*x + 2*c))/(((b^2*cos(c)^2 + b^2*sin(c)^
2)*d^2*x + (a*b*cos(c)^2 + a*b*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^2*x + (a*b*cos
(c)^2 + a*b*sin(c)^2)*d^2)*sin(d*x + c)^2)
Giac [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.35 (sec) , antiderivative size = 2205, normalized size of antiderivative = 22.27
\[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\text {Too large to display}
\]
[In]
integrate(x^2*sin(d*x+c)/(b*x+a),x, algorithm="giac")
[Out]
1/2*(a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d
^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*sin_
integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*real_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*t
an(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*
x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b^2*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^
2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^2*d^2*imag_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1
/2*c)^2*tan(1/2*c)^2 + 4*a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/
2*a*d/b) - 4*a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b) +
8*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b) - a^2*d^2*imag_part(c
os_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/
b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*t
an(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*imag_pa
rt(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*real_part(co
s_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*t
an(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 2*b^2*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^2*d^2*real_part(cos_int
egral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/
b) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*d*x*tan(1/2*d*x + 1/2
*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d^
2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2
+ a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2 - a^2*d^2*imag_part(cos_integral(-d*x -
a*d/b))*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2 - a^2*d^2*imag
_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2
*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 - 2*a*b*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 4*a^2*d^2*
imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*d^2*imag_part(cos_integral(-d*x - a*d/b
))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - a^2*d^2*ima
g_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d
/b)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*a*b*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/
b)^2 + 2*a*b*d*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*b^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^
2*d*x*tan(1/2*d*x + 1/2*c)^2 + 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^2*d^2*real_part
(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*b^2*d*x*tan(1/2*c)^2 - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/
b))*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) - 2*b^2*d*x*tan(1/2*a*d/b)
^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b)) - a^2*d^2*imag_part(cos_integral(-d*x - a*d/b)) + 2*a^2*d^2*
sin_integral((b*d*x + a*d)/b) - 2*a*b*d*tan(1/2*d*x + 1/2*c)^2 + 2*a*b*d*tan(1/2*c)^2 + 4*b^2*tan(1/2*d*x + 1/
2*c)*tan(1/2*c)^2 + 2*a*b*d*tan(1/2*a*d/b)^2 + 4*b^2*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d*x + 2*a*b
*d + 4*b^2*tan(1/2*d*x + 1/2*c))/(b^3*d^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^3*d^2*tan(1
/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + b^3*d^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + b^3*d^2*tan(1/2*c)^2*tan(1/
2*a*d/b)^2 + b^3*d^2*tan(1/2*d*x + 1/2*c)^2 + b^3*d^2*tan(1/2*c)^2 + b^3*d^2*tan(1/2*a*d/b)^2 + b^3*d^2)
Mupad [F(-1)]
Timed out. \[
\int \frac {x^2 \sin (c+d x)}{a+b x} \, dx=\int \frac {x^2\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x
\]
[In]
int((x^2*sin(c + d*x))/(a + b*x),x)
[Out]
int((x^2*sin(c + d*x))/(a + b*x), x)