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

Optimal result

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {a} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\sqrt {a} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )-4 b \cos (d x) \left (-\frac {2 \cos (c)}{d}+\frac {a x \sin (c)}{a+b x^2}\right )-4 b \left (\frac {a x \cos (c)}{a+b x^2}+\frac {2 \sin (c)}{d}\right ) \sin (d x)}{8 b^3} \] Input:

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

Output:

-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])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] + (- 
3*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) + Sqrt[ 
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] + Sq 
rt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) - 4*b*Cos[d*x]*((-2*C 
os[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
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3824, 3826, 2009, 3827, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3824

\(\displaystyle \frac {3 \int \frac {x^2 \sin (c+d x)}{b x^2+a}dx}{2 b}+\frac {d \int \frac {x^3 \cos (c+d x)}{b x^2+a}dx}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 3826

\(\displaystyle \frac {3 \int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b \left (b x^2+a\right )}\right )dx}{2 b}+\frac {d \int \frac {x^3 \cos (c+d x)}{b x^2+a}dx}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \int \frac {x^3 \cos (c+d x)}{b x^2+a}dx}{2 b}+\frac {3 \left (-\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\right )}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 3827

\(\displaystyle \frac {d \int \left (\frac {x \cos (c+d x)}{b}-\frac {a x \cos (c+d x)}{b \left (b x^2+a\right )}\right )dx}{2 b}+\frac {3 \left (-\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\right )}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (-\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\right )}{2 b}+\frac {d \left (-\frac {a \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {a \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {a \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\cos (c+d x)}{b d^2}+\frac {x \sin (c+d x)}{b d}\right )}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3824

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym 
bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))) 
, x] + (-Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*(a + b*x^n)^(p + 1) 
*Sin[c + d*x], x], x] - Simp[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 3826

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

rule 3827

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.25 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.18

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {{\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {3 \sqrt {a b}\, {\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}-\frac {3 \sqrt {a b}\, {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}+\frac {\operatorname {expIntegral}_{1}\left (-\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} a d}{8 b^{3}}+\frac {\operatorname {expIntegral}_{1}\left (-\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} a d}{8 b^{3}}-\frac {3 \,\operatorname {expIntegral}_{1}\left (-\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}\, {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}}}{8 b^{3}}+\frac {3 \,\operatorname {expIntegral}_{1}\left (-\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}\, {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}}}{8 b^{3}}-\frac {\cos \left (d x +c \right )}{b^{2} d}+\frac {d^{2} a x \sin \left (d x +c \right )}{2 b^{2} \left (x^{2} d^{2} b +d^{2} a \right )}\)	\(532\)
derivativedivides	\(\text {Expression too large to display}\)	\(3411\)
default	\(\text {Expression too large to display}\)	\(3411\)

Input:

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

Output:

1/8/b^3*exp((I*b*c+d*(a*b)^(1/2))/b)*Ei(1,(I*b*c+d*(a*b)^(1/2)-b*(I*d*x+I* 
c))/b)*a*d+1/8/b^3*exp((I*b*c-d*(a*b)^(1/2))/b)*Ei(1,(I*b*c-d*(a*b)^(1/2)- 
b*(I*d*x+I*c))/b)*a*d+3/8/b^3*(a*b)^(1/2)*exp((I*b*c+d*(a*b)^(1/2))/b)*Ei( 
1,(I*b*c+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)-3/8/b^3*(a*b)^(1/2)*exp((I*b*c-d* 
(a*b)^(1/2))/b)*Ei(1,(I*b*c-d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)+1/8/b^3*Ei(1,- 
(I*b*c+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp(-(I*b*c+d*(a*b)^(1/2))/b)*a*d+1 
/8/b^3*Ei(1,-(I*b*c-d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp(-(I*b*c-d*(a*b)^(1 
/2))/b)*a*d-3/8/b^3*Ei(1,-(I*b*c+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*(a*b)^(1/ 
2)*exp(-(I*b*c+d*(a*b)^(1/2))/b)+3/8/b^3*Ei(1,-(I*b*c-d*(a*b)^(1/2)-b*(I*d 
*x+I*c))/b)*(a*b)^(1/2)*exp(-(I*b*c-d*(a*b)^(1/2))/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 = 0.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {4 \, a b d x \sin \left (d x + c\right ) - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (b^{2} x^{2} + a b\right )} \cos \left (d x + c\right )}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}} \] Input:

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

Output:

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)*s 
qrt(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)*sqr 
t(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

-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*cos(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*si 
n(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)*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*co 
s(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(...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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