3.66 \(\int \frac {x^3 \sin (c+d x)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=431 \[ \frac {\sqrt {-a} d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sin (c+d x)}{2 b^2} \]

[Out]

1/2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/b^2+1/2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)
^(1/2)/b^(1/2))/b^2+1/2*sin(d*x+c)/b^2-1/2*x^2*sin(d*x+c)/b/(b*x^2+a)+1/2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d
*(-a)^(1/2)/b^(1/2))/b^2+1/2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/b^2-1/4*d*Ci(d*x+d*(-a)
^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+1/4*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a
)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+1/4*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/
b^(5/2)-1/4*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)

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Rubi [A]  time = 0.66, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3343, 3345, 3303, 3299, 3302, 3346, 2637, 3334} \[ \frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\sqrt {-a} d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sin (c+d x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

(Sqrt[-a]*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (Sqrt[-a]*d*C
os[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (CosIntegral[(Sqrt[-a]*d)/
Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt
[-a]*d)/Sqrt[b]])/(2*b^2) + Sin[c + d*x]/(2*b^2) - (x^2*Sin[c + d*x])/(2*b*(a + b*x^2)) - (Cos[c + (Sqrt[-a]*d
)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) + (Sqrt[-a]*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinInt
egral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[
b] + d*x])/(2*b^2) + (Sqrt[-a]*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(
5/2))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 3334

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])

Rule 3343

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

Rule 3345

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]

Rule 3346

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[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 {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx &=-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x \sin (c+d x)}{a+b x^2} \, dx}{b}+\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^2} \, dx}{2 b}\\ &=-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\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}{b}+\frac {d \int \left (\frac {\cos (c+d x)}{b}-\frac {a \cos (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {d \int \cos (c+d x) \, dx}{2 b^2}-\frac {(a d) \int \frac {\cos (c+d x)}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {(a d) \int \left (\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b^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}{2 b^{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}{2 b^{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}{2 b^{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}{2 b^{3/2}}\\ &=\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}-\frac {\left (\sqrt {-a} d\right ) \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} d\right ) \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2}\\ &=\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}-\frac {\left (\sqrt {-a} 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} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} 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} x} \, dx}{4 b^2}+\frac {\left (\sqrt {-a} 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} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} 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} x} \, dx}{4 b^2}\\ &=\frac {\sqrt {-a} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.89, size = 583, normalized size = 1.35 \[ \frac {i a^{3/2} d \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+i a^{3/2} d \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+2 b^{3/2} x^2 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-2 b^{3/2} x^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (a+b x^2\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (2 \sqrt {b} \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-i \sqrt {a} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+\left (a+b x^2\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (2 \sqrt {b} \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+i \sqrt {a} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+i \sqrt {a} b d x^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+i \sqrt {a} b d x^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+2 a \sqrt {b} \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-2 a \sqrt {b} \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+2 a \sqrt {b} \sin (c+d x)}{4 b^{5/2} \left (a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^3*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

((a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*((-I)*Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + 2*Sqrt[
b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]) + (a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(I*Sqrt[a]*d*Cos[
c + (I*Sqrt[a]*d)/Sqrt[b]] + 2*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) + 2*a*Sqrt[b]*Sin[c + d*x] + 2*a*Sqrt[b
]*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 2*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d
)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d
*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt
[b] + x)] - 2*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - 2*b^(3/2)*x^
2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/
Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegr
al[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)*(a + b*x^2))

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fricas [C]  time = 0.76, size = 291, normalized size = 0.68 \[ \frac {{\left (-4 i \, b x^{2} + 2 \, {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 4 i \, a\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-4 i \, b x^{2} + 2 \, {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 4 i \, a\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (4 i \, b x^{2} + 2 \, {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 4 i \, a\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (4 i \, b x^{2} + 2 \, {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 4 i \, a\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + 8 \, a \sin \left (d x + c\right )}{16 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/16*((-4*I*b*x^2 + 2*(-I*b*x^2 - I*a)*sqrt(a*d^2/b) - 4*I*a)*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)
) + (-4*I*b*x^2 + 2*(I*b*x^2 + I*a)*sqrt(a*d^2/b) - 4*I*a)*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) +
(4*I*b*x^2 + 2*(I*b*x^2 + I*a)*sqrt(a*d^2/b) + 4*I*a)*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (
4*I*b*x^2 + 2*(-I*b*x^2 - I*a)*sqrt(a*d^2/b) + 4*I*a)*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) + 8*
a*sin(d*x + c))/(b^3*x^2 + a*b^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^3*sin(d*x + c)/(b*x^2 + a)^2, x)

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maple [B]  time = 0.11, size = 2563, normalized size = 5.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sin(d*x+c)/(b*x^2+a)^2,x)

[Out]

1/d^4*(sin(d*x+c)*(-1/2*c*d^2*(3*a*d^2-b*c^2)/a/b*(d*x+c)+1/2*d^2*(a^2*d^4-b^2*c^4)/a/b^2)/((d*x+c)^2*b-2*(d*x
+c)*b*c+a*d^2+b*c^2)+1/4*d^2*(2*(d*(-a*b)^(1/2)+c*b)/b*a*d^2+a*c*d^2+b*c^3)/a/b^2/((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*d^2*(-2*(d*(-a*b)^(1/2)-c*b)/b*a*d^2+a*c*d^2+b*c^3)/a/b^2/(-(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*d^2*(3*(d*(-a*b)^(1/2)+c*b)*a*c*d^2-(d*(-a*b)^(1/2)+c*b)*b*c^3-a^2*d^4+b^2*c^4)/a/b^3/((d*(-a*b)^(1
/2)+c*b)/b-c)*(-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*d^2*(-3*(d*(-a*b)^(1/2)-c*b)*a*c*d^2+(d*(-a*b)^(1/2)-c*b)*b*c^3-a^2*d^4+b^2*c
^4)/a/b^3/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(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))+sin(d*x+c)*(3/2*c*d^2*(a*d^2-b*c^2)/a/b*(d*x+c)+3/2*c^2*d^
2*(a*d^2+b*c^2)/a/b)/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)-3/4*c*d^2*(a*d^2+b*c^2)/a/b^2/((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))-3/4*c*d^2*(a*d^2+b*c^2)/a/b^2/(-(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))-3/4*c*d^2*((d*(-
a*b)^(1/2)+c*b)/b*a*d^2-(d*(-a*b)^(1/2)+c*b)*c^2+a*c*d^2+b*c^3)/a/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(-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)
)-3/4*c*d^2*(-(d*(-a*b)^(1/2)-c*b)/b*a*d^2+(d*(-a*b)^(1/2)-c*b)*c^2+a*c*d^2+b*c^3)/a/b^2/(-(d*(-a*b)^(1/2)-c*b
)/b-c)*(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))+sin(d*x+c)*(3/2*c^3*d^2/a*(d*x+c)-3/2*c^2*d^2*(a*d^2+b*c^2)/a/b)/((d*x+c)^2*b-2*(d*x+c)*b
*c+a*d^2+b*c^2)+3/4*c^3*d^2/a/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))+3/4*c^3*d^2/a/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))-3/4*c^2*d^2*((d*(-a*b)^(1/2)+c*b)*c-a*d^2-b*c^2)/a/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(-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))-3/4*c^2*d^2*(-(d*(-a*b)^(1/2)-c*b)*c-a*d^2-b*c^2)/a/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(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))-c^3*
d^4*(sin(d*x+c)*(1/2/a/d^2*(d*x+c)-1/2*c/a/d^2)/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)+1/4/a/d^2/((d*(-a*b)^(
1/2)+c*b)/b-c)/b*(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/(-(d*(-a*b)^(1/2)-c*b)/b-c)/b*(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(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*sin(c + d*x))/(a + b*x^2)^2,x)

[Out]

int((x^3*sin(c + d*x))/(a + b*x^2)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sin {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*sin(d*x+c)/(b*x**2+a)**2,x)

[Out]

Integral(x**3*sin(c + d*x)/(a + b*x**2)**2, x)

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