\(\int \frac {x \sin (c+d x)}{(a+b x^3)^3} \, dx\) [111]

Optimal result

Integrand size = 17, antiderivative size = 1141 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {d \cos (c+d x)}{18 a b^2 x^3}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {2 d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {2 \operatorname {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {2 (-1)^{2/3} \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d^2 \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}+\frac {2 \sqrt [3]{-1} \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 (-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac {\sqrt [3]{-1} d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac {2 d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {2 \sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b} \]

[Out]

-2/27*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^2/b-2/27*d*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*c
os(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^2/b-2/27*d*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*cos(c-(-1)^(2/3)*a^(1/3)*
d/b^(1/3))/a^2/b+1/18*d*cos(d*x+c)/a/b^2/x^3-1/18*d*cos(d*x+c)/b^2/x^3/(b*x^3+a)-2/27*(-1)^(2/3)*cos(c+(-1)^(1
/3)*a^(1/3)*d/b^(1/3))*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)-1/54*(-1)^(1/3)*d^2*cos(c+(-1)^(1
/3)*a^(1/3)*d/b^(1/3))*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(4/3)-2/27*cos(c-a^(1/3)*d/b^(1/3))*Si(
a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)+1/54*d^2*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(
4/3)+2/27*(-1)^(1/3)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)+
1/54*(-1)^(2/3)*d^2*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(4/3)-2
/27*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/54*d^2*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(
c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)+2/27*d*Si(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^2/b-2/27*(-1)
^(2/3)*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)-1/54*(-1)^(1/3
)*d^2*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)+2/27*d*Si(-(-1)
^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^2/b+2/27*(-1)^(1/3)*Ci((-1)^(2/3)*a^(1/3)*
d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/54*(-1)^(2/3)*d^2*Ci((-1)^(2/3)*a^(1/3)*d
/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)+2/27*d*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*
sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^2/b-1/18*sin(d*x+c)/a/b^2/x^4+2/9*sin(d*x+c)/a^2/b/x-1/6*sin(d*x+c)/b/x/
(b*x^3+a)^2+1/18*sin(d*x+c)/b^2/x^4/(b*x^3+a)

Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 1141, normalized size of antiderivative = 1.00, number of steps used = 89, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3424, 3426, 3378, 3384, 3380, 3383, 3427, 3425, 3414} \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {\operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\sqrt [3]{-1} \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {\sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\cos (c+d x) d}{18 b^2 x^3 \left (b x^3+a\right )}+\frac {\cos (c+d x) d}{18 a b^2 x^3}-\frac {2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{27 a^2 b}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{27 a^2 b}+\frac {2 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}+\frac {2 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {2 (-1)^{2/3} \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sqrt [3]{-1} \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sin (c+d x)}{9 a^2 b x}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (b x^3+a\right )}-\frac {\sin (c+d x)}{6 b x \left (b x^3+a\right )^2}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 (-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}} \]

[In]

Int[(x*Sin[c + d*x])/(a + b*x^3)^3,x]

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x^3) - (d*Cos[c + d*x])/(18*b^2*x^3*(a + b*x^3)) - (2*d*Cos[c + ((-1)^(1/3)*a^(1/3)
*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*d*Cos[c - (a^(1/3)*d)/b^(1/3)]
*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral
[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3
)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)]
)/(54*a^(5/3)*b^(4/3)) - (2*(-1)^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a
^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) - ((-1)^(1/3)*d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*S
in[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) + (2*(-1)^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d
)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) + ((-1)^(2/3)*d^2*CosIntegral[(
(-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) - Sin[c + d
*x]/(18*a*b^2*x^4) + (2*Sin[c + d*x])/(9*a^2*b*x) - Sin[c + d*x]/(6*b*x*(a + b*x^3)^2) + Sin[c + d*x]/(18*b^2*
x^4*(a + b*x^3)) + (2*(-1)^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^
(1/3) - d*x])/(27*a^(7/3)*b^(2/3)) + ((-1)^(1/3)*d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)
^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(5/3)*b^(4/3)) - (2*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinInteg
ral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d
)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d
*x])/(54*a^(5/3)*b^(4/3)) + (2*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*
b) + (2*(-1)^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/
(27*a^(7/3)*b^(2/3)) + ((-1)^(2/3)*d^2*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)
*d)/b^(1/3) + d*x])/(54*a^(5/3)*b^(4/3)) + (2*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3
)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*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])

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 3425

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m - n + 1)*(a + b*x
^n)^(p + 1)*(Cos[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)*Cos[c + d*x], x], x] + Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[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 {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}-\frac {\int \frac {\sin (c+d x)}{x^2 \left (a+b x^3\right )^2} \, dx}{6 b}+\frac {d \int \frac {\cos (c+d x)}{x \left (a+b x^3\right )^2} \, dx}{6 b} \\ & = -\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \frac {\sin (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {d \int \frac {\cos (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac {d \int \frac {\cos (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{6 b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{18 b^2} \\ & = -\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \left (\frac {\sin (c+d x)}{a x^5}-\frac {b \sin (c+d x)}{a^2 x^2}+\frac {b^2 x \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^4}-\frac {b \cos (c+d x)}{a^2 x}+\frac {b^2 x^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^4}-\frac {b \cos (c+d x)}{a^2 x}+\frac {b^2 x^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{a x^3}-\frac {b \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2} \\ & = -\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \frac {x \sin (c+d x)}{a+b x^3} \, dx}{9 a^2}+\frac {2 \int \frac {\sin (c+d x)}{x^5} \, dx}{9 a b^2}-\frac {2 \int \frac {\sin (c+d x)}{x^2} \, dx}{9 a^2 b}-\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^4} \, dx}{18 a b^2}-\frac {d \int \frac {\cos (c+d x)}{x^4} \, dx}{6 a b^2}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{18 a^2 b}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{6 a^2 b}-\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{18 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{a+b x^3} \, dx}{18 a b} \\ & = \frac {2 d \cos (c+d x)}{27 a b^2 x^3}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {d^2 \sin (c+d x)}{36 a b^2 x^2}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \left (-\frac {\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac {d \int \left (\frac {\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}-\frac {d \int \left (\frac {\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^4} \, dx}{18 a b^2}-\frac {(2 d) \int \frac {\cos (c+d x)}{x} \, dx}{9 a^2 b}+\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{54 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{18 a b^2}+\frac {d^2 \int \left (-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{36 a b^2}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{6 a^2 b}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{6 a^2 b} \\ & = \frac {d \cos (c+d x)}{18 a b^2 x^3}+\frac {d^3 \cos (c+d x)}{36 a b^2 x}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac {2 d \cos (c) \operatorname {CosIntegral}(d x)}{9 a^2 b}-\frac {\sin (c+d x)}{18 a b^2 x^4}-\frac {d^2 \sin (c+d x)}{108 a b^2 x^2}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {2 d \sin (c) \text {Si}(d x)}{9 a^2 b}-\frac {2 \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 \sqrt [3]{-1}\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 (-1)^{2/3}\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{54 a b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{108 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{36 a b^2}+\frac {d^4 \int \frac {\sin (c+d x)}{x} \, dx}{36 a b^2}-\frac {(2 d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{9 a^2 b}+\frac {(2 d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{9 a^2 b} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.41 (sec) , antiderivative size = 698, normalized size of antiderivative = 0.61 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-i a d^2 \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-a d^2 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-a d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+i a d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-4 i b \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-4 b \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+4 i b \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+4 b d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 i b d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2-4 i b d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 b d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {i a d^2 \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-a d^2 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-a d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-i a d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+4 i b \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-4 b \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-4 i b \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+4 b d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}^2+4 i b d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2+4 i b d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 b d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]-\frac {6 b \cos (d x) \left (a d \left (a+b x^3\right ) \cos (c)+b x^2 \left (7 a+4 b x^3\right ) \sin (c)\right )}{\left (a+b x^3\right )^2}-\frac {6 b \left (b x^2 \left (7 a+4 b x^3\right ) \cos (c)-a d \left (a+b x^3\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^3\right )^2}}{108 a^2 b^2} \]

[In]

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

[Out]

-1/108*(RootSum[a + b*#1^3 & , ((-I)*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1
)]*Sin[c + d*#1] - a*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]
- (4*I)*b*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c
+ d*#1]*SinIntegral[d*(x - #1)]*#1 + (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*C
osIntegral[d*(x - #1)]*#1^2 - (4*I)*b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - (4*I)*b*d*Cos[c + d*#1]*S
inIntegral[d*(x - #1)]*#1^2 - 4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + RootSum[a + b*#1^3
& , (I*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - a*d^2*Cos[c
+ d*#1]*SinIntegral[d*(x - #1)] - I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + (4*I)*b*Cos[c + d*#1]*CosIn
tegral[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#1]*SinIntegral[d*(x - #1
)]*#1 - (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 +
(4*I)*b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + (4*I)*b*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 -
4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - (6*b*Cos[d*x]*(a*d*(a + b*x^3)*Cos[c] + b*x^2*(7*
a + 4*b*x^3)*Sin[c]))/(a + b*x^3)^2 - (6*b*(b*x^2*(7*a + 4*b*x^3)*Cos[c] - a*d*(a + b*x^3)*Sin[c])*Sin[d*x])/(
a + b*x^3)^2)/(a^2*b^2)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.38 (sec) , antiderivative size = 610, normalized size of antiderivative = 0.53

method	result	size
risch	\(\frac {i d c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}+6 i c -6 \textit {\_R1} +10\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}+\frac {i d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (-2 i b \textit {\_R1} \,c^{2}+\textit {\_R1}^{2} b c +4 i b \,\textit {\_R1}^{2}+a \,d^{3}-c^{3} b +2 i b \,c^{2}+2 b c \textit {\_R1} -4 i \textit {\_R1} b +6 c b \right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{108 a^{2} b^{2}}-\frac {i d c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}-6 i c +6 \textit {\_R1} +10\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}-\frac {i d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (-2 i b \textit {\_R1} \,c^{2}+\textit {\_R1}^{2} b c -4 i b \,\textit {\_R1}^{2}+a \,d^{3}-c^{3} b -2 i b \,c^{2}-2 b c \textit {\_R1} -4 i \textit {\_R1} b +6 c b \right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (i d x +i c -\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{108 a^{2} b^{2}}+\frac {d^{4} \left (d^{3} x^{3} b +a \,d^{3}\right ) \cos \left (d x +c \right )}{18 a b \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d \left (-4 b \,x^{5} d^{5}-7 a \,d^{5} x^{2}\right ) \sin \left (d x +c \right )}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\)	\(610\)
derivativedivides	\(\text {Expression too large to display}\)	\(847\)
default	\(\text {Expression too large to display}\)	\(847\)

[In]

int(x*sin(d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/108*I*d/a^2/b*c*sum((-2*I*c*_R1+6*I*c+_R1^2-c^2-6*_R1+10)/(-2*I*c*_R1+_R1^2-c^2)*exp(_R1)*Ei(1,_R1-I*d*x-I*c
),_R1=RootOf(-3*I*_Z^2*b*c-I*d^3*a+I*b*c^3+b*_Z^3-3*c^2*b*_Z))+1/108*I*d/a^2/b^2*sum((-2*I*b*_R1*c^2+4*I*b*_R1
^2+2*I*b*c^2+_R1^2*b*c+a*d^3-c^3*b-4*I*_R1*b+2*b*c*_R1+6*c*b)/(2*I*c*_R1-_R1^2+c^2)*exp(_R1)*Ei(1,_R1-I*d*x-I*
c),_R1=RootOf(-3*I*_Z^2*b*c-I*d^3*a+I*b*c^3+b*_Z^3-3*c^2*b*_Z))-1/108*I*d/a^2/b*c*sum((-2*I*c*_R1-6*I*c+_R1^2-
c^2+6*_R1+10)/(-2*I*c*_R1+_R1^2-c^2)*exp(-_R1)*Ei(1,I*d*x+I*c-_R1),_R1=RootOf(-3*I*_Z^2*b*c-I*d^3*a+I*b*c^3+b*
_Z^3-3*c^2*b*_Z))-1/108*I*d/a^2/b^2*sum((-2*I*b*_R1*c^2-4*I*b*_R1^2-2*I*b*c^2+_R1^2*b*c+a*d^3-c^3*b-4*I*_R1*b-
2*b*c*_R1+6*c*b)/(2*I*c*_R1-_R1^2+c^2)*exp(-_R1)*Ei(1,I*d*x+I*c-_R1),_R1=RootOf(-3*I*_Z^2*b*c-I*d^3*a+I*b*c^3+
b*_Z^3-3*c^2*b*_Z))+1/18*d^4/a*(b*d^3*x^3+a*d^3)/b/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*cos(d*x+c)-1/18*d*(-4*b
*d^5*x^5-7*a*d^5*x^2)/a^2/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*sin(d*x+c)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 1319, normalized size of antiderivative = 1.16 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/216*((8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b + sqrt(3)*(b^
3*x^6 + 2*a*b^2*x^3 + a^2*b))*(I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(I*a*b^
2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3))*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) -
1))*e^(1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) - I*c) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(-I
*b^3*x^6 - 2*I*a*b^2*x^3 - I*a^2*b - sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(-I*a*d^3/b)^(2/3) - (a*b^2*d^3*
x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3))*(-I*a*d^3/b)^(1/3
))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) + I*c) + (8*a
*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b - sqrt(3)*(b^3*x^6 + 2*a*
b^2*x^3 + a^2*b))*(I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(-I*a*b^2*d^3*x^6 -
2*I*a^2*b*d^3*x^3 - I*a^3*d^3))*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*
(I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) - I*c) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(-I*b^3*x^6 -
2*I*a*b^2*x^3 - I*a^2*b + sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(-I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2
*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3))*(-I*a*d^3/b)^(1/3))*Ei(I*d*
x + 1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) + I*c) + 2*(4*a*b^2*d^3
*x^6 + 8*a^2*b*d^3*x^3 + 4*a^3*d^3 + 4*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b)*(-I*a*d^3/b)^(2/3) + (a*b^2*d^3*x
^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)
) + 2*(4*a*b^2*d^3*x^6 + 8*a^2*b*d^3*x^3 + 4*a^3*d^3 + 4*(-I*b^3*x^6 - 2*I*a*b^2*x^3 - I*a^2*b)*(I*a*d^3/b)^(2
/3) + (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c -
(I*a*d^3/b)^(1/3)) - 12*(a^2*b*d^3*x^3 + a^3*d^3)*cos(d*x + c) - 12*(4*a*b^2*d^2*x^5 + 7*a^2*b*d^2*x^2)*sin(d*
x + c))/(a^3*b^3*d^2*x^6 + 2*a^4*b^2*d^2*x^3 + a^5*b*d^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

-1/2*((cos(c)^2 + sin(c)^2)*x*cos(d*x + c) + (x*cos(d*x + c)^2*cos(c) + x*cos(c)*sin(d*x + c)^2)*cos(d*x + 2*c
) + 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 +
a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9
+ 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3
*sin(c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*(8*b*x^3 - a)*cos(d*x + c)/(b^4*d*x^12 + 4*a*b^3*d*x^9 + 6*a^2*b^2
*d*x^6 + 4*a^3*b*d*x^3 + a^4*d), x) + 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(
c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((
b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin
(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*(8*b*x^3 - a)*cos(d*x + c)/((b^4
*d*x^12 + 4*a*b^3*d*x^9 + 6*a^2*b^2*d*x^6 + 4*a^3*b*d*x^3 + a^4*d)*cos(d*x + c)^2 + (b^4*d*x^12 + 4*a*b^3*d*x^
9 + 6*a^2*b^2*d*x^6 + 4*a^3*b*d*x^3 + a^4*d)*sin(d*x + c)^2), x) + (x*cos(d*x + c)^2*sin(c) + x*sin(d*x + c)^2
*sin(c))*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3
*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((b^3*cos(c)^2 +
b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^3 +
(a^3*cos(c)^2 + a^3*sin(c)^2)*d)*sin(d*x + c)^2)

Giac [F]

\[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x\,\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]

[In]

int((x*sin(c + d*x))/(a + b*x^3)^3,x)

[Out]

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