3.1.62 \(\int \frac {\sin (c+d x)}{x (a+b x^2)} \, dx\) [62]

Optimal. Leaf size=197 \[ \frac {\text {Ci}(d x) \sin (c)}{a}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}+\frac {\cos (c) \text {Si}(d x)}{a}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3426, 3384, 3380, 3383} \begin {gather*} -\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a}-\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}+\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}+\frac {\sin (c) \text {CosIntegral}(d x)}{a}+\frac {\cos (c) \text {Si}(d x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^2)),x]

[Out]

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

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 3426

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 179, normalized size = 0.91 \begin {gather*} -\frac {-2 \text {Ci}(d x) \sin (c)+\text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )+\text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )-2 \cos (c) \text {Si}(d x)+\cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(-2*CosIntegral[d*x]*Sin[c] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + C
osIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - 2*Cos[c]*SinIntegral[d*x] + Cos[c
- (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral
[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/a

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Maple [A]
time = 0.09, size = 200, normalized size = 1.02

method	result	size
derivativedivides	\(-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 a}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 a}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a}\)	\(200\)
default	\(-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 a}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 a}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a}\)	\(200\)
risch	\(-\frac {i {\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a}-\frac {i {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a}+\frac {i {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a}+\frac {i {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a}+\frac {i {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a}-\frac {{\mathrm e}^{-i c} \pi \,\mathrm {csgn}\left (d x \right )}{2 a}+\frac {{\mathrm e}^{-i c} \sinIntegral \left (d x \right )}{a}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, -i d x \right )}{2 a}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.38, size = 168, normalized size = 0.85 \begin {gather*} \frac {-2 i \, {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 2 i \, {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + i \, {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - i \, {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )}}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{x\,\left (b\,x^2+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x*(a + b*x^2)),x)

[Out]

int(sin(c + d*x)/(x*(a + b*x^2)), x)

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