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

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

[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])

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Rubi [A]  time = 0.536306, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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])

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 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]

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*}

Mathematica [C]  time = 0.321835, size = 172, normalized size = 0.81 \[ \frac{i \left (\sin \left (a-\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )-\sin \left (a+\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\cos \left (a-\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{Si}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\cos \left (a+\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{i b \sqrt{c}}{\sqrt{d}}-b x\right )\right )}{2 \sqrt{c} \sqrt{d}} \]

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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Maple [A]  time = 0.022, size = 229, normalized size = 1.1 \begin{align*} b \left ({\frac{1}{2\,d} \left ({\it Si} \left ( bx+a-{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) \cos \left ({\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) +{\it Ci} \left ( bx+a-{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) \sin \left ({\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) } \right ) \right ) \left ({\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }-a \right ) ^{-1}}+{\frac{1}{2\,d} \left ({\it Si} \left ( bx+a+{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) \cos \left ({\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) -{\it Ci} \left ( bx+a+{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) \sin \left ({\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) } \right ) \right ) \left ( -{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }-a \right ) ^{-1}} \right ) \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )}{d x^{2} + c}\,{d x} \end{align*}

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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Fricas [C]  time = 2.13078, size = 377, normalized size = 1.77 \begin{align*} \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} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b x \right )}}{c + d x^{2}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )}{d x^{2} + c}\,{d x} \end{align*}

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)