\(\int \frac {x^4 \sin (c+d x)}{(a+b x^2)^2} \, dx\) [65]

Optimal result

Integrand size = 19, antiderivative size = 450 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\cos (c+d x)}{b^2 d}-\frac {a d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {3 \sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {x \sin (c+d x)}{2 b^2}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {-a} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {a d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {a d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3} \]

[Out]

-cos(d*x+c)/b^2/d-1/4*a*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/b^3-1/4*a*d*Ci(-d*x+d*(-a)^
(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^(1/2))/b^3+1/2*x*sin(d*x+c)/b^2-1/2*x^3*sin(d*x+c)/b/(b*x^2+a)+1/4*a*d*Si(
d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/b^3+1/4*a*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1
/2)/b^(1/2))/b^3+3/4*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)-3/4*cos(c-d*(
-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)-3/4*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-
a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+3/4*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/
b^(5/2)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3424, 3426, 2718, 3414, 3384, 3380, 3383, 3427, 3377} \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {3 \sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {3 \sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {a d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}+\frac {a d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {x \sin (c+d x)}{2 b^2}-\frac {\cos (c+d x)}{b^2 d} \]

[In]

Int[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

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

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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])

Rule 3424

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 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]

Rule 3427

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*} \text {integral}& = -\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \frac {x^2 \sin (c+d x)}{a+b x^2} \, dx}{2 b}+\frac {d \int \frac {x^3 \cos (c+d x)}{a+b x^2} \, dx}{2 b} \\ & = -\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}+\frac {d \int \left (\frac {x \cos (c+d x)}{b}-\frac {a x \cos (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b} \\ & = -\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \sin (c+d x) \, dx}{2 b^2}-\frac {(3 a) \int \frac {\sin (c+d x)}{a+b x^2} \, dx}{2 b^2}+\frac {d \int x \cos (c+d x) \, dx}{2 b^2}-\frac {(a d) \int \frac {x \cos (c+d x)}{a+b x^2} \, dx}{2 b^2} \\ & = -\frac {3 \cos (c+d x)}{2 b^2 d}+\frac {x \sin (c+d x)}{2 b^2}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\int \sin (c+d x) \, dx}{2 b^2}-\frac {(3 a) \int \left (\frac {\sqrt {-a} \sin (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sin (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b^2}-\frac {(a d) \int \left (-\frac {\cos (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b^2} \\ & = -\frac {\cos (c+d x)}{b^2 d}+\frac {x \sin (c+d x)}{2 b^2}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\left (3 \sqrt {-a}\right ) \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (3 \sqrt {-a}\right ) \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2}+\frac {(a d) \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^{5/2}}-\frac {(a d) \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^{5/2}} \\ & = -\frac {\cos (c+d x)}{b^2 d}+\frac {x \sin (c+d x)}{2 b^2}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\left (3 \sqrt {-a} \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}{4 b^2}-\frac {\left (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^{5/2}}+\frac {\left (3 \sqrt {-a} \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}{4 b^2}+\frac {\left (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^{5/2}}-\frac {\left (3 \sqrt {-a} \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}{4 b^2}+\frac {\left (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^{5/2}}-\frac {\left (3 \sqrt {-a} \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}{4 b^2}+\frac {\left (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^{5/2}} \\ & = -\frac {\cos (c+d x)}{b^2 d}-\frac {a d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {3 \sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {x \sin (c+d x)}{2 b^2}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {-a} \cos \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 {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^3}-\frac {3 \sqrt {-a} \cos \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 {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^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {a} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\sqrt {a} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )-4 b \cos (d x) \left (-\frac {2 \cos (c)}{d}+\frac {a x \sin (c)}{a+b x^2}\right )-4 b \left (\frac {a x \cos (c)}{a+b x^2}+\frac {2 \sin (c)}{d}\right ) \sin (d x)}{8 b^3} \]

[In]

Integrate[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

-1/8*(Sqrt[a]*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*((3*Sqrt[b] + Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralE
i[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] + (-3*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) + Sqr
t[a]*E^(I*c - (Sqrt[a]*d)/Sqrt[b])*((3*Sqrt[b] + Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]
*d)/Sqrt[b]) + I*d*x] + (-3*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) - 4*b*Cos[d*x]*((
-2*Cos[c])/d + (a*x*Sin[c])/(a + b*x^2)) - 4*b*((a*x*Cos[c])/(a + b*x^2) + (2*Sin[c])/d)*Sin[d*x])/b^3

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.16

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {3 \sqrt {a b}\, {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}-\frac {3 \sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}+\frac {{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}-\frac {3 \sqrt {a b}\, {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}+\frac {3 \sqrt {a b}\, {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}-\frac {\cos \left (d x +c \right )}{b^{2} d}+\frac {d^{2} a x \sin \left (d x +c \right )}{2 b^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )}\)	\(524\)
derivativedivides	\(\text {Expression too large to display}\)	\(3411\)
default	\(\text {Expression too large to display}\)	\(3411\)

[In]

int(x^4*sin(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/8/b^3*exp((I*c*b+d*(a*b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*a*d+1/8/b^3*exp((I*c*b-d*(a*b
)^(1/2))/b)*Ei(1,-(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)*a*d+3/8/b^3*(a*b)^(1/2)*exp((I*c*b+d*(a*b)^(1/2))/b)
*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)-3/8/b^3*(a*b)^(1/2)*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(-I*c*b+d*
(a*b)^(1/2)+b*(I*d*x+I*c))/b)+1/8/b^3*exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/
b)*a*d+1/8/b^3*exp(-(I*c*b-d*(a*b)^(1/2))/b)*Ei(1,(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)*a*d-3/8/b^3*(a*b)^(1
/2)*exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)+3/8/b^3*(a*b)^(1/2)*exp(-(I*c*b
-d*(a*b)^(1/2))/b)*Ei(1,(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)-cos(d*x+c)/b^2/d+1/2*d^2*a*x/b^2/(b*d^2*x^2+a*
d^2)*sin(d*x+c)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {4 \, a b d x \sin \left (d x + c\right ) - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (b^{2} x^{2} + a b\right )} \cos \left (d x + c\right )}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}} \]

[In]

integrate(x^4*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 + 3*(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 - 3*(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 + 3*(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 - 3*(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)) - 8*(b^2*x^2 + a*b)*cos(d*x + c))/(b^4*d*x^2 + a*b^3*d)

Sympy [F]

\[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \sin {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]

[In]

integrate(x**4*sin(d*x+c)/(b*x**2+a)**2,x)

[Out]

Integral(x**4*sin(c + d*x)/(a + b*x**2)**2, x)

Maxima [F]

\[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^4*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*((b*cos(c)^2 + b*sin(c)^2)*d*x^4*cos(d*x + c) - 4*(a*cos(c)^2 + a*sin(c)^2)*x*sin(d*x + c) + ((b*d*x^4*co
s(c) + 4*a*x*sin(c))*cos(d*x + c)^2 + (b*d*x^4*cos(c) + 4*a*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) - 2*(((b^
3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin
(c)^2)*d^2)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*
x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrate(-2*(a^2*d*x*cos(d*x + c) - (3*a*b*x^2 -
a^2)*sin(d*x + c))/(b^4*d^2*x^6 + 3*a*b^3*d^2*x^4 + 3*a^2*b^2*d^2*x^2 + a^3*b*d^2), x) - 2*(((b^3*cos(c)^2 +
b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)*c
os(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*
cos(c)^2 + a^2*b*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrate(-2*(a^2*d*x*cos(d*x + c) - (3*a*b*x^2 - a^2)*sin(d*x
+ c))/((b^4*d^2*x^6 + 3*a*b^3*d^2*x^4 + 3*a^2*b^2*d^2*x^2 + a^3*b*d^2)*cos(d*x + c)^2 + (b^4*d^2*x^6 + 3*a*b^
3*d^2*x^4 + 3*a^2*b^2*d^2*x^2 + a^3*b*d^2)*sin(d*x + c)^2), x) + ((b*d*x^4*sin(c) - 4*a*x*cos(c))*cos(d*x + c)
^2 + (b*d*x^4*sin(c) - 4*a*x*cos(c))*sin(d*x + c)^2)*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 +
2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^3*c
os(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)
^2)*d^2)*sin(d*x + c)^2)

Giac [F]

\[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^4*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^4*sin(d*x + c)/(b*x^2 + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

int((x^4*sin(c + d*x))/(a + b*x^2)^2,x)

[Out]

int((x^4*sin(c + d*x))/(a + b*x^2)^2, x)