3.1.69 \(\int \frac {\sin (c+d x)}{(a+b x^2)^2} \, dx\) [69]

Optimal. Leaf size=476 \[ -\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}+\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b} \]

[Out]

-1/4*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/a/b-1/4*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+
d*(-a)^(1/2)/b^(1/2))/a/b+1/4*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a/b+1/4*d*Si(d*x-d*(-
a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/a/b-1/4*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))
/(-a)^(3/2)/b^(1/2)+1/4*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)+1/4*Ci(d*x
+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)-1/4*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+
d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)-1/4*sin(d*x+c)/a/b^(1/2)/((-a)^(1/2)-x*b^(1/2))+1/4*sin(d*x+c)/a/b^(1
/2)/((-a)^(1/2)+x*b^(1/2))

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Rubi [A]
time = 0.59, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3414, 3378, 3384, 3380, 3383} \begin {gather*} -\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}-\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}+\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*x^2)^2,x]

[Out]

-1/4*(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(a*b) - (d*Cos[c - (Sqrt[-a]*d)
/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c -
(Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/
Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - Sin[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(4*a*Sqrt[
b]*(Sqrt[-a] + Sqrt[b]*x)) + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(
3/2)*Sqrt[b]) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Cos[c - (
Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (d*Sin[c - (Sqrt[-a]*d)
/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b)

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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 3414

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 (c+d x)}{\left (a+b x^2\right )^2} \, dx &=\int \left (-\frac {b \sin (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sin (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \sin (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx\\ &=-\frac {b \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a}-\frac {b \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a}-\frac {b \int \frac {\sin (c+d x)}{-a b-b^2 x^2} \, dx}{2 a}\\ &=-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {b \int \left (-\frac {\sqrt {-a} \sin (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \sin (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 a}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a}-\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a}\\ &=-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{3/2}}+\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{3/2}}-\frac {\left (d \cos \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}+b x} \, dx}{4 a}+\frac {\left (d \cos \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}-b x} \, dx}{4 a}+\frac {\left (d \sin \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}+b x} \, dx}{4 a}+\frac {\left (d \sin \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}-b x} \, dx}{4 a}\\ &=-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{3/2}}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{3/2}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{3/2}}+\frac {\sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{3/2}}\\ &=-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}+\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.45, size = 585, normalized size = 1.23 \begin {gather*} \frac {-\left (\left (a+b x^2\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \left (\sqrt {a} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-i \sqrt {b} \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )\right )-\left (a+b x^2\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \left (\sqrt {a} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+i \sqrt {b} \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+2 \sqrt {a} b x \sin (c+d x)+i a \sqrt {b} \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 )+i b^{3/2} x^2 \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 )+a^{3/2} d \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\sqrt {a} b d x^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i a \sqrt {b} \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+i b^{3/2} x^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-a^{3/2} d \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\sqrt {a} b d x^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{4 a^{3/2} b \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] - I*Sqrt[b]*
Sin[c - (I*Sqrt[a]*d)/Sqrt[b]])) - (a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c +
(I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) + 2*Sqrt[a]*b*x*Sin[c + d*x] + I*a*Sqrt[b]*
Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/
Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I
*Sqrt[a])/Sqrt[b] + x)] + Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] +
x)] + I*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*b^(3/2)*x^2*Cos[
c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]
*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt
[a]*d)/Sqrt[b] - d*x])/(4*a^(3/2)*b*(a + b*x^2))

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Maple [A]
time = 0.06, size = 491, normalized size = 1.03

method	result	size
derivativedivides	\(d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{2 a \,d^{2}}-\frac {c}{2 a \,d^{2}}\right )}{d^{2} a +b \,c^{2}-2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}}-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{4 a \,d^{2} b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{4 a \,d^{2} b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {-\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{4 a b \,d^{2}}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{4 a b \,d^{2}}\right )\)	\(491\)
default	\(d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{2 a \,d^{2}}-\frac {c}{2 a \,d^{2}}\right )}{d^{2} a +b \,c^{2}-2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}}-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{4 a \,d^{2} b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{4 a \,d^{2} b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {-\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{4 a b \,d^{2}}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{4 a b \,d^{2}}\right )\)	\(491\)
risch	\(\frac {d \,{\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}+\frac {d \,{\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}-\frac {{\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{8 a^{2} b}+\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}+\frac {d \,{\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}+\frac {d \,{\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}+\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}-\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}-\frac {d^{2} x \sin \left (d x +c \right )}{2 \left (-2 i \left (i d x +i c \right ) b c +\left (i d x +i c \right )^{2} b -d^{2} a -b \,c^{2}\right ) a}\)	\(565\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^3*(sin(d*x+c)*(1/2/a/d^2*(d*x+c)-1/2*c/a/d^2)/(d^2*a+b*c^2-2*b*c*(d*x+c)+b*(d*x+c)^2)-1/4/a/d^2/b/(-(d*(-a*b
)^(1/2)+c*b)/b+c)*(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))-1/4/a/d^2/b/((d*(-a*b)^(1/2)-c*b)/b+c)*(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))-1/4/a/b/d^2*(-Si(d*x+c-(d
*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b)
)-1/4/a/b/d^2*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*c
os((d*(-a*b)^(1/2)-c*b)/b)))

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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)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.38, size = 333, normalized size = 0.70 \begin {gather*} \frac {4 \, a b d x \sin \left (d x + c\right ) - {\left (a b d^{2} x^{2} + a^{2} d^{2} - {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )}}{8 \, {\left (a^{2} b^{2} d x^{2} + a^{3} b d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*a*b*d*x*sin(d*x + c) - (a*b*d^2*x^2 + a^2*d^2 - (b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b)
)*e^(I*c + sqrt(a*d^2/b)) - (a*b*d^2*x^2 + a^2*d^2 + (b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/b))*
e^(I*c - sqrt(a*d^2/b)) - (a*b*d^2*x^2 + a^2*d^2 - (b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d^2/b))*e
^(-I*c + sqrt(a*d^2/b)) - (a*b*d^2*x^2 + a^2*d^2 + (b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b))*e
^(-I*c - sqrt(a*d^2/b)))/(a^2*b^2*d*x^2 + a^3*b*d)

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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 )}}{\left (a + b x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(a + b*x^2)^2,x)

[Out]

int(sin(c + d*x)/(a + b*x^2)^2, x)

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