3.31 \(\int \frac {\sin (a+b x)}{c+d x^2} \, dx\)

Optimal. Leaf size=213 \[ -\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Ci}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Ci}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

[Out]

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

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Rubi [A]  time = 0.54, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3333, 3303, 3299, 3302} \[ -\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]/(c + d*x^2),x]

[Out]

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

Rule 3299

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 3302

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 3303

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 3333

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[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*} \int \frac {\sin (a+b x)}{c+d x^2} \, dx &=\int \left (\frac {\sqrt {-c} \sin (a+b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \sin (a+b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sin (a+b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\sin (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 {\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 ) \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 {\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 {\cos \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}\\ &=-\frac {\text {Ci}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right ) \sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Ci}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right ) \sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \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 {\cos \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*}

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Mathematica [C]  time = 0.32, size = 172, normalized size = 0.81 \[ \frac {i \left (\sin \left (a-\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Ci}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )-\sin \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Ci}\left (b \left (x-\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )+\cos \left (a-\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Si}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )+\cos \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {i b \sqrt {c}}{\sqrt {d}}-b x\right )\right )}{2 \sqrt {c} \sqrt {d}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[a + b*x]/(c + d*x^2),x]

[Out]

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

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fricas [C]  time = 0.56, size = 187, normalized size = 0.88 \[ \frac {\sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} c}{d}}\right )} + \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} c}{d}}\right )}}{4 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 229, normalized size = 1.08 \[ b \left (\frac {\Si \left (b x +a -\frac {b \sqrt {-c d}+d a}{d}\right ) \cos \left (\frac {b \sqrt {-c d}+d a}{d}\right )+\Ci \left (b x +a -\frac {b \sqrt {-c d}+d a}{d}\right ) \sin \left (\frac {b \sqrt {-c d}+d a}{d}\right )}{2 \left (\frac {b \sqrt {-c d}+d a}{d}-a \right ) d}+\frac {\Si \left (b x +a +\frac {b \sqrt {-c d}-d a}{d}\right ) \cos \left (\frac {b \sqrt {-c d}-d a}{d}\right )-\Ci \left (b x +a +\frac {b \sqrt {-c d}-d a}{d}\right ) \sin \left (\frac {b \sqrt {-c d}-d a}{d}\right )}{2 \left (-\frac {b \sqrt {-c d}-d a}{d}-a \right ) d}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (a+b\,x\right )}{d\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)/(c + d*x^2),x)

[Out]

int(sin(a + b*x)/(c + d*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (a + b x \right )}}{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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