\(\int \frac {\cos (a+b x)}{c+d x^2} \, dx\) [46]

Optimal result

Integrand size = 16, antiderivative size = 213 \[ \int \frac {\cos (a+b x)}{c+d x^2} \, dx=\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3415, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x)}{c+d x^2} \, dx=\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \operatorname {CosIntegral}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

[In]

Int[Cos[a + b*x]/(c + d*x^2),x]

[Out]

(Cos[a + (b*Sqrt[-c])/Sqrt[d]]*CosIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a - (b*Sqr
t[-c])/Sqrt[d]]*CosIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d]) + (Sin[a + (b*Sqrt[-c])/Sqrt[d]]
*SinIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) + (Sin[a - (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b
*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])

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 3415

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.77 \[ \int \frac {\cos (a+b x)}{c+d x^2} \, dx=\frac {i e^{-i a-\frac {b \sqrt {c}}{\sqrt {d}}} \left (-e^{\frac {2 b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \sqrt {c}}{\sqrt {d}}-i b x\right )+\operatorname {ExpIntegralEi}\left (\frac {b \sqrt {c}}{\sqrt {d}}-i b x\right )+e^{2 i a} \left (e^{\frac {2 b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \sqrt {c}}{\sqrt {d}}+i b x\right )-\operatorname {ExpIntegralEi}\left (\frac {b \sqrt {c}}{\sqrt {d}}+i b x\right )\right )\right )}{4 \sqrt {c} \sqrt {d}} \]

[In]

Integrate[Cos[a + b*x]/(c + d*x^2),x]

[Out]

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

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(b \left (-\frac {\operatorname {Si}\left (-x b -a +\frac {b \sqrt {-c d}+a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}+a d}{d}\right )+\operatorname {Ci}\left (x b +a -\frac {b \sqrt {-c d}+a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}+a d}{d}\right )}{2 d \left (-\frac {b \sqrt {-c d}+a d}{d}+a \right )}-\frac {-\operatorname {Si}\left (-x b -a -\frac {b \sqrt {-c d}-a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}-a d}{d}\right )+\operatorname {Ci}\left (x b +a +\frac {b \sqrt {-c d}-a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}-a d}{d}\right )}{2 d \left (\frac {b \sqrt {-c d}-a d}{d}+a \right )}\right )\)	\(231\)
default	\(b \left (-\frac {\operatorname {Si}\left (-x b -a +\frac {b \sqrt {-c d}+a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}+a d}{d}\right )+\operatorname {Ci}\left (x b +a -\frac {b \sqrt {-c d}+a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}+a d}{d}\right )}{2 d \left (-\frac {b \sqrt {-c d}+a d}{d}+a \right )}-\frac {-\operatorname {Si}\left (-x b -a -\frac {b \sqrt {-c d}-a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}-a d}{d}\right )+\operatorname {Ci}\left (x b +a +\frac {b \sqrt {-c d}-a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}-a d}{d}\right )}{2 d \left (\frac {b \sqrt {-c d}-a d}{d}+a \right )}\right )\)	\(231\)
risch	\(-\frac {i \sqrt {c d}\, {\mathrm e}^{-\frac {i a d +b \sqrt {c d}}{d}} \operatorname {Ei}_{1}\left (-\frac {i a d +b \sqrt {c d}-\left (i b x +i a \right ) d}{d}\right )}{4 c d}-\frac {i \sqrt {c d}\, {\mathrm e}^{\frac {i a d +b \sqrt {c d}}{d}} \operatorname {Ei}_{1}\left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a \right ) d}{d}\right )}{4 c d}+\frac {i \sqrt {c d}\, {\mathrm e}^{\frac {i a d -b \sqrt {c d}}{d}} \operatorname {Ei}_{1}\left (-\frac {-i a d +b \sqrt {c d}+\left (i b x +i a \right ) d}{d}\right )}{4 c d}+\frac {i \sqrt {c d}\, {\mathrm e}^{-\frac {i a d -b \sqrt {c d}}{d}} \operatorname {Ei}_{1}\left (\frac {-i a d +b \sqrt {c d}+\left (i b x +i a \right ) d}{d}\right )}{4 c d}\)	\(262\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (a+b x)}{c+d x^2} \, dx=-\frac {-i \, \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} + i \, \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} c}{d}}\right )} + i \, \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - i \, \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} c}{d}}\right )}}{4 \, b c} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cos(a + b*x)/(c + d*x^2),x)

[Out]

int(cos(a + b*x)/(c + d*x^2), x)