\(\int \frac {\sin (a+b x)}{c+d x+e x^2} \, dx\) [32]

Optimal result

Integrand size = 19, antiderivative size = 271 \[ \int \frac {\sin (a+b x)}{c+d x+e x^2} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sin \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}-\frac {\operatorname {CosIntegral}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sin \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}+\frac {\cos \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cos \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]

[Out]

cos(a-1/2*b*(d-(-4*c*e+d^2)^(1/2))/e)*Si(b*x+1/2*b*(d-(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)-cos(a-1/2*b*(d
+(-4*c*e+d^2)^(1/2))/e)*Si(b*x+1/2*b*(d+(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)+Ci(b*x+1/2*b*(d-(-4*c*e+d^2)
^(1/2))/e)*sin(a-1/2*b*(d-(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)-Ci(b*x+1/2*b*(d+(-4*c*e+d^2)^(1/2))/e)*sin
(a-1/2*b*(d+(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6860, 3384, 3380, 3383} \[ \int \frac {\sin (a+b x)}{c+d x+e x^2} \, dx=\frac {\sin \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \operatorname {CosIntegral}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\sin \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \operatorname {CosIntegral}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}+\frac {\cos \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cos \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]

[In]

Int[Sin[a + b*x]/(c + d*x + e*x^2),x]

[Out]

(CosIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sin[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4
*c*e] - (CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sin[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt
[d^2 - 4*c*e] + (Cos[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*
x])/Sqrt[d^2 - 4*c*e] - (Cos[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2
*e) + b*x])/Sqrt[d^2 - 4*c*e]

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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 e \sin (a+b x)}{\sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}+2 e x\right )}-\frac {2 e \sin (a+b x)}{\sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}+2 e x\right )}\right ) \, dx \\ & = \frac {(2 e) \int \frac {\sin (a+b x)}{d-\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}-\frac {(2 e) \int \frac {\sin (a+b x)}{d+\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}} \\ & = \frac {\left (2 e \cos \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\sin \left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d-\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}-\frac {\left (2 e \cos \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\sin \left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d+\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}+\frac {\left (2 e \sin \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\cos \left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d-\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}-\frac {\left (2 e \sin \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\cos \left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d+\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}} \\ & = \frac {\operatorname {CosIntegral}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sin \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}-\frac {\operatorname {CosIntegral}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sin \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}+\frac {\cos \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cos \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Si}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.65 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.90 \[ \int \frac {\sin (a+b x)}{c+d x+e x^2} \, dx=\frac {i e^{-\frac {1}{2} i \left (2 a+\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{e}\right )} \left (e^{\frac {i b d}{e}} \operatorname {ExpIntegralEi}\left (-\frac {i b \left (d-\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )-e^{i \left (2 a+\frac {b \sqrt {d^2-4 c e}}{e}\right )} \operatorname {ExpIntegralEi}\left (\frac {i b \left (d-\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )-e^{\frac {i b \left (d+\sqrt {d^2-4 c e}\right )}{e}} \operatorname {ExpIntegralEi}\left (-\frac {i b \left (d+\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )+e^{2 i a} \operatorname {ExpIntegralEi}\left (\frac {i b \left (d+\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )\right )}{2 \sqrt {d^2-4 c e}} \]

[In]

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

[Out]

((I/2)*(E^((I*b*d)/e)*ExpIntegralEi[((-1/2*I)*b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/e] - E^(I*(2*a + (b*Sqrt[d^2
- 4*c*e])/e))*ExpIntegralEi[((I/2)*b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/e] - E^((I*b*(d + Sqrt[d^2 - 4*c*e]))/e)
*ExpIntegralEi[((-1/2*I)*b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/e] + E^((2*I)*a)*ExpIntegralEi[((I/2)*b*(d + Sqrt[
d^2 - 4*c*e] + 2*e*x))/e]))/(Sqrt[d^2 - 4*c*e]*E^((I/2)*(2*a + (b*(d + Sqrt[d^2 - 4*c*e]))/e)))

Maple [A] (verified)

Time = 1.67 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(b \left (\frac {-\operatorname {Si}\left (-x b -a +\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \cos \left (\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )+\operatorname {Ci}\left (x b +a -\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \sin \left (\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{\sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {-\operatorname {Si}\left (-x b -a -\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \cos \left (\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )-\operatorname {Ci}\left (x b +a +\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \sin \left (\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{\sqrt {-4 b^{2} c e +b^{2} d^{2}}}\right )\)	\(328\)
default	\(b \left (\frac {-\operatorname {Si}\left (-x b -a +\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \cos \left (\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )+\operatorname {Ci}\left (x b +a -\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \sin \left (\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{\sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {-\operatorname {Si}\left (-x b -a -\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \cos \left (\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )-\operatorname {Ci}\left (x b +a +\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right ) \sin \left (\frac {-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{\sqrt {-4 b^{2} c e +b^{2} d^{2}}}\right )\)	\(328\)
risch	\(\frac {\sqrt {4 b^{2} c e -b^{2} d^{2}}\, \operatorname {Ei}_{1}\left (\frac {2 i a e -i b d -2 e \left (i b x +i a \right )-\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}\right ) {\mathrm e}^{\frac {2 i a e -i b d -\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}}}{2 b \left (4 c e -d^{2}\right )}-\frac {\sqrt {4 b^{2} c e -b^{2} d^{2}}\, \operatorname {Ei}_{1}\left (\frac {2 i a e -i b d -2 e \left (i b x +i a \right )+\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}\right ) {\mathrm e}^{\frac {2 i a e -i b d +\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}}}{2 b \left (4 c e -d^{2}\right )}+\frac {\sqrt {4 b^{2} c e -b^{2} d^{2}}\, \operatorname {Ei}_{1}\left (-\frac {2 i a e -i b d -2 e \left (i b x +i a \right )+\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}\right ) {\mathrm e}^{-\frac {2 i a e -i b d +\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}}}{2 b \left (4 c e -d^{2}\right )}-\frac {\sqrt {4 b^{2} c e -b^{2} d^{2}}\, \operatorname {Ei}_{1}\left (-\frac {2 i a e -i b d -2 e \left (i b x +i a \right )-\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}\right ) {\mathrm e}^{-\frac {2 i a e -i b d -\sqrt {4 b^{2} c e -b^{2} d^{2}}}{2 e}}}{2 b \left (4 c e -d^{2}\right )}\)	\(486\)

[In]

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

[Out]

b*(1/(-4*b^2*c*e+b^2*d^2)^(1/2)*(-Si(-x*b-a+1/2/e*(2*a*e-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2)))*cos(1/2/e*(2*a*e-b*d
+(-4*b^2*c*e+b^2*d^2)^(1/2)))+Ci(x*b+a-1/2/e*(2*a*e-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2)))*sin(1/2/e*(2*a*e-b*d+(-4*
b^2*c*e+b^2*d^2)^(1/2))))-1/(-4*b^2*c*e+b^2*d^2)^(1/2)*(-Si(-x*b-a-1/2*(-2*a*e+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))
/e)*cos(1/2*(-2*a*e+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)-Ci(x*b+a+1/2*(-2*a*e+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)
*sin(1/2*(-2*a*e+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.60 \[ \int \frac {\sin (a+b x)}{c+d x+e x^2} \, dx=-\frac {e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {-2 i \, b e x - i \, b d - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {i \, b d - 2 i \, a e + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {-2 i \, b e x - i \, b d + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {i \, b d - 2 i \, a e - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 i \, b e x + i \, b d - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {-i \, b d + 2 i \, a e + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )} - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 i \, b e x + i \, b d + e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) e^{\left (\frac {-i \, b d + 2 i \, a e - e \sqrt {-\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )}}{2 \, {\left (b d^{2} - 4 \, b c e\right )}} \]

[In]

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

[Out]

-1/2*(e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(1/2*(-2*I*b*e*x - I*b*d - e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2))/e)*e
^(1/2*(I*b*d - 2*I*a*e + e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2))/e) - e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(1/2*(-
2*I*b*e*x - I*b*d + e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2))/e)*e^(1/2*(I*b*d - 2*I*a*e - e*sqrt(-(b^2*d^2 - 4*b^2*
c*e)/e^2))/e) + e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(1/2*(2*I*b*e*x + I*b*d - e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e
^2))/e)*e^(1/2*(-I*b*d + 2*I*a*e + e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2))/e) - e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2)
*Ei(1/2*(2*I*b*e*x + I*b*d + e*sqrt(-(b^2*d^2 - 4*b^2*c*e)/e^2))/e)*e^(1/2*(-I*b*d + 2*I*a*e - e*sqrt(-(b^2*d^
2 - 4*b^2*c*e)/e^2))/e))/(b*d^2 - 4*b*c*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sin(a + b*x)/(c + d*x + e*x^2),x)

[Out]

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